🧠:[<think>Okay, so I need to find the fourth-order derivative of y = sin x. Hmm, derivatives can get a bit tricky when you go to higher orders, but let me think step by step. First, I remember that the derivative of sin x is cos x. That's the first derivative. Then the second derivative would be the derivative of cos x, right? And the derivative of cos x is -sin x. So the second derivative is -sin x. Moving on to the third derivative, that's the derivative of -sin x. The derivative of sin x is cos x, so with the negative sign, it should be -cos x. Wait, let me check that again. If the second derivative is -sin x, then taking the derivative of that would be -cos x. Yeah, that's correct.Now, the fourth derivative would be the derivative of -cos x. The derivative of cos x is -sin x, so multiplying by the negative sign already there, it becomes -(-sin x) which is sin x. Oh, so the fourth derivative brings us back to sin x? That makes sense because I remember something about trigonometric functions having cyclical derivatives. Like, each time you take the derivative of sin or cos, it cycles through sin, cos, -sin, -cos, and then back to sin. Let me verify that cycle:First derivative: cos xSecond derivative: -sin xThird derivative: -cos xFourth derivative: sin xYep, exactly. So after four derivatives, we end up back at the original function. So the fourth-order derivative of sin x is sin x itself. That seems right. But let me make sure I didn't make a mistake with the signs. Let's go through each step again:1st derivative: d/dx [sin x] = cos x2nd derivative: d/dx [cos x] = -sin x3rd derivative: d/dx [-sin x] = -cos x4th derivative: d/dx [-cos x] = sin xYes, each time the sign changes appropriately. So the fourth derivative is indeed sin x. I think that's correct. Another way to think about it is that the derivatives of sin x are periodic with period 4. So every fourth derivative will loop back to sin x. Therefore, regardless of the order, if it's a multiple of 4, the derivative is sin x. Since 4 is the fourth derivative, that's exactly the case here.Just to double-check, maybe I can recall a formula for the nth derivative of sin x. I think it's sin(x + nπ/2). Let me test that. For n=1, sin(x + π/2) = cos x, which matches. For n=2, sin(x + π) = -sin x, which is the second derivative. For n=3, sin(x + 3π/2) = -cos x, correct again. And for n=4, sin(x + 2π) = sin x, since 2π is a full rotation. Yep, that formula works. So applying it for n=4 gives sin x, which matches what I found earlier. Therefore, I'm confident that the fourth-order derivative of y = sin x is y itself, sin x. No mistakes in the steps, and the formula for the nth derivative confirms it. So the answer should be sin x.Final AnswerThe fourth-order derivative of the function is boxed{sin x}.</think>To find the fourth-order derivative of the function ( y = sin x ), we proceed step by step:1. First derivative: [ frac{d}{dx} (sin x) = cos x ]2. Second derivative: [ frac{d}{dx} (cos x) = -sin x ]3. Third derivative: [ frac{d}{dx} (-sin x) = -cos x ]4. Fourth derivative: [ frac{d}{dx} (-cos x) = sin x ]Additionally, using the formula for the ( n )-th derivative of ( sin x ), which is ( sin left( x + frac{npi}{2} right) ), for ( n = 4 ):[sin left( x + frac{4pi}{2} right) = sin (x + 2pi) = sin x]Thus, the fourth-order derivative of the function ( y = sin x ) is (boxed{sin x}).,<think>Okay, so I need to find the fourth-order derivative of y = sin x. Hmm, derivatives can get a bit tricky when you go to higher orders, but let's take it step by step. Let me remember how to take derivatives of sine functions.First, the original function is y = sin x. The first derivative of sin x is cos x. I think that's right. Wait, yes, because the derivative of sin x is cos x. Then the second derivative would be the derivative of cos x. What's the derivative of cos x? Oh, right, it's -sin x. So the second derivative is -sin x.Now, moving on to the third derivative. That's the derivative of -sin x. The derivative of sin x is cos x, so the derivative of -sin x would be -cos x. So third derivative is -cos x. Then the fourth derivative. That's the derivative of -cos x. The derivative of cos x is -sin x, so multiplying by the negative sign already there, the derivative becomes -(-sin x) which is sin x. Wait, so the fourth derivative brings us back to sin x?Let me check that again to be sure. First derivative: cos x. Second derivative: -sin x. Third derivative: -cos x. Fourth derivative: sin x. Yes, that seems to cycle every four derivatives. So after four derivatives, we end up back at the original function. That's interesting. So for y = sin x, the fourth-order derivative is the same as the original function. Therefore, the answer should be sin x. But let me confirm each step again just to make sure I didn't mix up any signs.First derivative: d/dx sin x = cos x. Correct. Second derivative: d/dx cos x = -sin x. Right. Third derivative: d/dx (-sin x) = -cos x. Yes. Fourth derivative: d/dx (-cos x) = -(-sin x) = sin x. Yep, that checks out. So each time you take the derivative, you either get cos x, -sin x, -cos x, sin x, and then it repeats. So every four derivatives, it cycles back. Therefore, the fourth derivative of sin x is indeed sin x. So the answer is y'''' = sin x. Alternatively, maybe there's another way to think about it. For functions like sin x and cos x, their derivatives cycle every four steps. So for any nth derivative, you can find n modulo 4 to determine which function it is. For example, first derivative (n=1) is cos x, n=2 is -sin x, n=3 is -cos x, n=4 is sin x, which brings us back. So yes, fourth derivative is sin x. I don't think I made any mistakes here, but let me write out each derivative explicitly:1. y = sin x2. y' = cos x3. y'' = -sin x4. y''' = -cos x5. y'''' = sin xYep, step by step, each derivative follows from the previous one correctly. The signs alternate, and the functions cycle between sin and cos. So after four derivatives, we're back to the original function. Therefore, the fourth-order derivative of y = sin x is sin x.Final AnswerThe fourth-order derivative of the function is boxed{sin x}.</think>To find the fourth-order derivative of the function ( y = sin x ), we proceed step by step:1. The first derivative of ( y = sin x ) is: [ y' = cos x ]2. The second derivative is the derivative of ( cos x ): [ y'' = -sin x ]3. The third derivative is the derivative of ( -sin x ): [ y''' = -cos x ]4. The fourth derivative is the derivative of ( -cos x ): [ y'''' = sin x ]Each derivative cycles through the functions (sin x), (cos x), (-sin x), (-cos x), and back to (sin x) every four derivatives. Therefore, the fourth-order derivative of ( y = sin x ) is:[boxed{sin x}]]

🔑:Solution. Differentiating successively, we determine:y^{prime}=cos x, y^{prime prime}=-sin x, y^{prime prime prime}=-cos x, y^{mathrm{IV}}=sin x

🧠:[<think>Okay, so I have this problem here about 100 books divided into different categories and types. Let me try to parse it step by step. First, the books are split into two categories: liberal arts and science. Then, each of those can be either hardcover or softcover. The total number of books is 100. The given numbers are: 40 liberal arts books, 70 hardcover books, and 20 softcover science books. The questions are about probabilities related to selecting certain types of books. Let me start by trying to make a table or a matrix to organize the information. That might help me visualize the different categories and types. So, let's think of a 2x2 table where rows are categories (liberal arts and science) and columns are types (hardcover and softcover). Then we can fill in the numbers we know and solve for the unknowns.So, the table would look like this:``` | Hardcover | Softcover | Total------------------------------------------------ Liberal Arts | ? | ? | 40 Science | ? | 20 | ?------------------------------------------------ Total | 70 | 30 | 100```Wait, the total number of hardcover books is 70, so softcover must be 100 - 70 = 30. And they mentioned there are 20 softcover science books, so that's filled in the science softcover cell. Now, the total number of science books would be the total of science hardcover plus 20 (softcover). But we don't know the total number of science books yet. However, since there are 40 liberal arts books, the total science books must be 100 - 40 = 60. So, updating the table:``` | Hardcover | Softcover | Total------------------------------------------------ Liberal Arts | ? | ? | 40 Science | ? | 20 | 60------------------------------------------------ Total | 70 | 30 | 100```Now, let's find the number of science hardcover books. Since science has 60 total books and 20 are softcover, then the hardcover science books must be 60 - 20 = 40. So, science hardcover is 40. Then, the total hardcover books are 70, so liberal arts hardcover would be total hardcover minus science hardcover: 70 - 40 = 30. Then, liberal arts softcover would be total liberal arts minus liberal arts hardcover: 40 - 30 = 10. Let me check if that adds up. Liberal arts hardcover 30, softcover 10, total 40. Science hardcover 40, softcover 20, total 60. Total hardcover 30 + 40 = 70, softcover 10 + 20 = 30. Total books 70 + 30 = 100. Yep, that checks out.So, filling in the table:``` | Hardcover | Softcover | Total------------------------------------------------ Liberal Arts | 30 | 10 | 40 Science | 40 | 20 | 60------------------------------------------------ Total | 70 | 30 | 100```Alright, now that the table is filled out, let's tackle the questions.Question (1): The probability of randomly selecting a liberal arts hardcover book. Probability is the number of favorable outcomes over the total number of outcomes. So, the number of liberal arts hardcover books is 30, total books is 100. So the probability should be 30/100, which simplifies to 3/10 or 0.3. Wait, let me make sure. The problem says "randomly selecting a liberal arts hardcover book." Yes, that's exactly the 30 books. So 30 out of 100. So the probability is 30/100 = 3/10. So the answer should be 3/10. Let me just double-check the numbers again. Yes, 30 liberal arts hardcover. 100 total. So that's correct.Question (2): The probability of first randomly selecting a liberal arts book, then returning it and randomly selecting a hardcover book.Okay, this is a two-step probability with replacement, since the first book is returned. So, the two events are independent. First, the probability of selecting a liberal arts book. There are 40 liberal arts books out of 100, so that's 40/100 = 2/5. Then, after returning it, the probability of selecting a hardcover book. There are 70 hardcover books out of 100, so 70/100 = 7/10. Since the events are independent, we multiply the probabilities: (2/5) * (7/10) = 14/50 = 7/25. Let me verify. First selection: 40/100 = 0.4. Second selection: 70/100 = 0.7. Multiply them: 0.4 * 0.7 = 0.28. 7/25 is 0.28. Correct. Wait, but just to make sure there isn't any trick here. Since we are selecting a liberal arts book first, and then a hardcover, but since we return the first book, the total remains 100 for both selections. So yes, independent events. So multiplying the two probabilities is the right approach.Hmm, seems straightforward. Let me check again. 40 liberal arts, so 40/100. 70 hardcover, so 70/100. Multiply them. Yep. So 40*70 = 2800. 100*100 = 10000. 2800/10000 = 28/100 = 7/25. Correct. Therefore, the answers are 3/10 and 7/25.Final Answer(1) The probability is boxed{dfrac{3}{10}}.(2) The probability is boxed{dfrac{7}{25}}.</think>Given 100 books divided into two categories (liberal arts and science) and two types (hardcover and softcover), we need to find the probabilities for two specific events. The known values are 40 liberal arts books, 70 hardcover books, and 20 softcover science books. We construct a table to organize the information:[begin{array}{|c|c|c|c|}hline & text{Hardcover} & text{Softcover} & text{Total} hlinetext{Liberal Arts} & 30 & 10 & 40 hlinetext{Science} & 40 & 20 & 60 hlinetext{Total} & 70 & 30 & 100 hlineend{array}](1) The number of liberal arts hardcover books is 30. Therefore, the probability of randomly selecting a liberal arts hardcover book is:[frac{30}{100} = frac{3}{10}](2) The probability of first selecting a liberal arts book (40 out of 100) and then, after returning it, selecting a hardcover book (70 out of 100) is calculated by multiplying the probabilities of these independent events:[left( frac{40}{100} right) times left( frac{70}{100} right) = frac{2}{5} times frac{7}{10} = frac{14}{50} = frac{7}{25}]Final answers:(1) boxed{dfrac{3}{10}} (2) boxed{dfrac{7}{25}},<think>Okay, let's see. I need to solve these two probability questions based on the given data about the books. Let me start by organizing the information provided.There are 100 books total. They are divided into two categories: liberal arts and science. Also, each book is either hardcover or softcover. So, we have two categories (liberal arts and science) and two types (hardcover and softcover). The problem gives us some numbers:- 40 liberal arts books- 70 hardcover books- 20 softcover science booksAnd we need to find:1. The probability of randomly selecting a liberal arts hardcover book.2. The probability of first selecting a liberal arts book, returning it, then selecting a hardcover book.Let me tackle the first problem first.Problem 1: Probability of selecting a liberal arts hardcover bookTo find this probability, I need to know how many liberal arts hardcover books there are, then divide that by the total number of books (100). The problem doesn't give this number directly, so I need to calculate it using the given data.Let me think. We know there are 40 liberal arts books in total. These 40 include both hardcover and softcover. Similarly, the total number of hardcover books is 70, which includes both liberal arts and science.We also know there are 20 softcover science books. Maybe we can use that to figure out the number of hardcover science books, and then work our way to the liberal arts hardcover books.Let me try to create a table to organize this. A contingency table might help here.Let's denote:- L: Liberal Arts- S: Science- H: Hardcover- C: Softcover (since S is taken for Science, let's use C for softcover)We need to fill in the table:| | Hardcover (H) | Softcover (C) | Total ||--------|---------------|---------------|-------|| Liberal Arts (L) | ? | ? | 40 || Science (S) | ? | 20 | ? || Total | 70 | ? | 100 |We know the total number of books is 100. The total hardcover is 70, so the total softcover must be 100 - 70 = 30. Therefore, the total softcover is 30. We already know that softcover science books are 20, so the remaining softcover books must be liberal arts. So, softcover liberal arts books would be 30 total softcover - 20 softcover science = 10.So updating the table:| | Hardcover (H) | Softcover (C) | Total ||--------|---------------|---------------|-------|| Liberal Arts (L) | ? | 10 | 40 || Science (S) | ? | 20 | ? || Total | 70 | 30 | 100 |Now, for the Liberal Arts row, since total liberal arts books are 40, and softcover is 10, then hardcover liberal arts must be 40 - 10 = 30. So:| | Hardcover (H) | Softcover (C) | Total ||--------|---------------|---------------|-------|| Liberal Arts (L) | 30 | 10 | 40 || Science (S) | ? | 20 | ? || Total | 70 | 30 | 100 |Now, let's look at the Science row. The total hardcover books are 70, and we already have 30 hardcover liberal arts, so hardcover science books must be 70 - 30 = 40.Then, the total science books would be hardcover science + softcover science = 40 + 20 = 60.So updating the table:| | Hardcover (H) | Softcover (C) | Total ||--------|---------------|---------------|-------|| Liberal Arts (L) | 30 | 10 | 40 || Science (S) | 40 | 20 | 60 || Total | 70 | 30 | 100 |Now, verifying the totals: Liberal Arts total is 40, Science is 60, which adds up to 100. Hardcover total is 70 (30+40), softcover is 30 (10+20), which also adds up to 100. Perfect, the table checks out.So, the number of liberal arts hardcover books is 30. Therefore, the probability is 30/100 = 0.3.Problem 2: Probability of first selecting a liberal arts book, returning it, then selecting a hardcover bookThis is a two-step experiment with replacement (since the book is returned after the first selection). The probability of both events happening in sequence is the product of their individual probabilities.First, the probability of selecting a liberal arts book is 40/100 = 0.4.After returning the book, the total number of books is still 100, so the probability of selecting a hardcover book is 70/100 = 0.7.Therefore, the combined probability is 0.4 * 0.7 = 0.28.Wait a second, is that correct? Let me make sure.Since we are returning the book, the two events are independent. The first selection doesn't affect the second selection. Therefore, yes, the probability is simply the product of the two individual probabilities.Alternatively, if we didn't return the book, the second probability would change, but since we are returning it, it remains the same. So, yes, 0.4 * 0.7 = 0.28.So, converting these to fractions might be better for the answer. Let's check:30/100 simplifies to 3/10. 0.3 is 3/10.For the second part, 40/100 is 2/5, 70/100 is 7/10. Multiplying them: 2/5 * 7/10 = 14/100 = 7/50, which is 0.14. Wait, wait, wait, hold on. Wait, 0.4 * 0.7 is 0.28, which is 28/100 = 7/25. Hmm, why the confusion here.Wait, 40/100 is 0.4, 70/100 is 0.7. 0.4*0.7 is 0.28, which is 28/100, which simplifies to 7/25. Wait, that contradicts my previous thought. Wait, 40/100 * 70/100 = (40*70)/(100*100) = 2800/10000 = 28/100 = 7/25. Yes, that's 0.28.Wait, earlier when I thought 2/5 *7/10, 2/5 is 40/100, and 7/10 is 70/100. So 2/5 *7/10 = (2*7)/(5*10)=14/50=7/25. Yes, same as before. So 7/25 is 0.28. So 7/25 is the correct fraction.Therefore, the answers should be 3/10 and 7/25.Wait, but let me just cross-verify everything once again. Maybe I made a mistake in the table.Total books: 100.Liberal arts: 40 (so Science is 60).Hardcover: 70 (so Softcover is 30).Softcover science: 20. Therefore, softcover liberal arts is total softcover (30) minus softcover science (20) = 10. Then, hardcover liberal arts is total liberal arts (40) minus softcover liberal arts (10) = 30. Then, hardcover science is total hardcover (70) minus hardcover liberal arts (30) = 40. Then, science total is 40 +20=60, which matches. So all numbers check out.Therefore, the number of liberal arts hardcover is 30, so probability is 30/100=3/10=0.3.Second problem: First, probability of liberal arts is 40/100=2/5. Then, probability of hardcover is 70/100=7/10. Since it's with replacement, independent. Multiply them: (2/5)*(7/10)=14/50=7/25=0.28.Yes, so both answers are correct.Final Answer(1) boxed{dfrac{3}{10}} (2) boxed{dfrac{7}{25}}</think>Given 100 books divided into two categories (liberal arts and science) and two types (hardcover and softcover), we need to find the probabilities for two scenarios:1. The probability of randomly selecting a liberal arts hardcover book.2. The probability of first selecting a liberal arts book, returning it, and then selecting a hardcover book.We start by organizing the information into a contingency table:| | Hardcover (H) | Softcover (C) | Total ||--------|---------------|---------------|-------|| Liberal Arts (L) | ? | 10 | 40 || Science (S) | 40 | 20 | 60 || Total | 70 | 30 | 100 |Key steps:1. Total softcover books: 100 - 70 = 30.2. Softcover science books: 20, so softcover liberal arts books = 30 - 20 = 10.3. Hardcover liberal arts books: 40 (total liberal arts) - 10 (softcover liberal arts) = 30.4. Hardcover science books: 70 (total hardcover) - 30 (hardcover liberal arts) = 40.5. Total science books: 40 (hardcover) + 20 (softcover) = 60.Calculations:1. Probability of selecting a liberal arts hardcover book: [ frac{text{Number of liberal arts hardcover books}}{text{Total number of books}} = frac{30}{100} = frac{3}{10} ]2. Probability of first selecting a liberal arts book, returning it, then selecting a hardcover book: - Probability of selecting a liberal arts book: (frac{40}{100} = frac{2}{5}) - Probability of selecting a hardcover book: (frac{70}{100} = frac{7}{10}) - Combined probability (independent events): [ frac{2}{5} times frac{7}{10} = frac{14}{50} = frac{7}{25} ]Final Answers:(1) boxed{dfrac{3}{10}} (2) boxed{dfrac{7}{25}}]

🔑:(1) The total number of events is 100, including 40 liberal arts books and 60 science books. There are 70 hardcover books, 20 softcover science books, and 40 hardcover liberal arts books. Therefore, there are 30 hardcover liberal arts books and 10 softcover liberal arts books. The probability of randomly selecting a liberal arts hardcover book is frac{30}{100} = boxed{0.3}.(2) The total number of events is 100 times 100. The probability P is calculated as follows: P = frac{C^1_{40} C^1_{70}}{100 times 100} = frac{4}{10} times frac{7}{10} = boxed{0.28}.

🧠:[<think>Okay, let me try to figure out this geometry problem. So we have a square ABCD with side length 1. Point P is on the diagonal AC, and AP is given as x. Then there's a point E on CD such that PE is perpendicular to PB. We need to find the relationship between the area y of the pentagon ABPED and x. Hmm, okay.First, let me draw a square ABCD. Let's label the square with A at the bottom left corner, B at the bottom right, C at the top right, and D at the top left. So the diagonal AC goes from A to C. Since the square has side length 1, all sides are 1 unit long.Point P is somewhere on AC, not at A or C. So AP = x, which means PC = AC - AP. Since the diagonal of a square with side length 1 is √2, AC = √2. Therefore, PC = √2 - x.Now, point E is on CD, and PE is perpendicular to PB. We need to find coordinates for all these points to express PE perpendicular to PB. Maybe coordinate geometry would help here.Let me set up a coordinate system. Let's place point A at (0,0), so B is at (1,0), C is at (1,1), and D is at (0,1). The diagonal AC goes from (0,0) to (1,1). So any point P on AC can be parameterized. Since AP = x, and AC is length √2, the coordinates of P would be (x/√2, x/√2). Wait, because moving along the diagonal AC from A, each unit along the diagonal corresponds to √2 in length. So if AP = x, then the coordinates of P are (x/√2, x/√2). Hmm, but maybe there's a simpler way to parameterize it.Alternatively, since AC is the line from (0,0) to (1,1), any point on AC can be written as (t, t) where t ranges from 0 to 1. Then AP is the distance from A to P, which would be √(t² + t²) = t√2. So if AP = x, then t = x / √2. Therefore, the coordinates of P are (x/√2, x/√2). That makes sense. So P is (x/√2, x/√2). Alternatively, maybe using t from 0 to 1 where t is the fraction along AC. But perhaps using x as given is better.Now, point E is on CD. CD is the top side from C (1,1) to D (0,1). Wait, no, CD is actually from C (1,1) to D (0,1) if the square is labeled in order. Wait, hold on: If ABCD is a square, going around, then the order is A, B, C, D. So AB is the bottom side, BC is the right side, CD is the top side, and DA is the left side. So CD goes from C (1,1) to D (0,1). Wait, that's correct. So CD is the top side, horizontal from (1,1) to (0,1). So point E is somewhere on CD, which is the line segment from (1,1) to (0,1). Therefore, the coordinates of E can be written as (s, 1), where s ranges from 0 to 1. So if we let E be (s,1), then s is between 0 and 1.Now, we need PE to be perpendicular to PB. Let's write the coordinates of P and B. P is (x/√2, x/√2), as established earlier. B is (1,0). So vector PB is B - P, which is (1 - x/√2, 0 - x/√2) = (1 - x/√2, -x/√2). Similarly, vector PE is E - P, which is (s - x/√2, 1 - x/√2). Since PE is perpendicular to PB, their dot product must be zero. Therefore:(1 - x/√2)(s - x/√2) + (-x/√2)(1 - x/√2) = 0.Let me expand this:(1 - x/√2)(s - x/√2) - (x/√2)(1 - x/√2) = 0.Factor out (1 - x/√2):(1 - x/√2)[(s - x/√2) - x/√2] = 0.But since P is not equal to A or C, x is between 0 and √2, so (1 - x/√2) might not necessarily be zero. Wait, actually, when x is between 0 and √2, x/√2 is between 0 and 1. So (1 - x/√2) could be positive or negative depending on x. But maybe we can proceed.Wait, expanding the original equation:First term: (1 - x/√2)(s - x/√2) = (1)(s) - (1)(x/√2) - (x/√2)(s) + (x/√2)^2Second term: -x/√2*(1 - x/√2) = -x/√2 + (x/√2)^2Adding both terms together:s - x/√2 - (x s)/√2 + (x²)/(2) - x/√2 + (x²)/2 = 0Combine like terms:s - (x/√2 + x/√2) - (x s)/√2 + (x²/2 + x²/2) = 0Simplify:s - (2x)/√2 - (x s)/√2 + x² = 0Simplify 2x/√2 = x√2, so:s - x√2 - (x s)/√2 + x² = 0Now, let's collect terms with s:s[1 - x/√2] - x√2 + x² = 0Therefore:s[1 - x/√2] = x√2 - x²Then:s = [x√2 - x²] / [1 - x/√2]Let me factor numerator and denominator:Numerator: x(√2 - x)Denominator: 1 - x/√2 = (√2 - x)/√2Therefore, s = [x(√2 - x)] / [(√2 - x)/√2] ] = x(√2 - x) * [√2 / (√2 - x)] ) = x√2Wait, that simplifies nicely. The (√2 - x) cancels out, so s = x√2 * √2 / √2? Wait, wait, let me recheck that step.Wait, denominator is (√2 - x)/√2. So when we divide by that, it's equivalent to multiplying by √2/(√2 - x). So:s = [x(√2 - x)] * [√2 / (√2 - x)] = x√2So s = x√2. But s is the x-coordinate of point E on CD. Since CD is from (1,1) to (0,1), s must be between 0 and 1. Therefore, x√2 must be between 0 and 1. But x is the length AP, which is between 0 and √2. Wait, but since P is not at A or C, x is between 0 and √2. However, s must be between 0 and 1 because E is on CD. Therefore, x√2 must be between 0 and 1. So x must be between 0 and 1/√2 ≈ 0.707. But the problem states that P is on AC different from A and C. Wait, but if x can go up to √2, but s = x√2 must be ≤ 1, so x ≤ 1/√2. That seems contradictory. Wait, is there a mistake here?Wait, maybe my parameterization is wrong. Let me check.Earlier, we said that AP = x. Since AC is the diagonal of the square with length √2, x can range from 0 to √2. But if s = x√2 must be ≤1 (since E is on CD, s is between 0 and1), then x must be ≤1/√2. So this suggests that x is between 0 and 1/√2. But the problem states that P is on AC different from A and C. So maybe the problem allows x up to √2, but in that case, s would have to be greater than 1, which is impossible. Therefore, perhaps there's a mistake in my calculation.Wait, let's go back. When we set up the coordinates for P, we assumed that AP = x, so we parameterized P as (x/√2, x/√2). But perhaps using a different parameterization would be better. Let's instead parameterize P with a parameter t, where t is the x-coordinate of P. Since the diagonal AC is the line y = x, so if P is (t, t), then AP is the distance from (0,0) to (t, t), which is √(t² + t²) = t√2. Therefore, if AP = x, then t = x / √2. Therefore, coordinates of P are (x/√2, x/√2). So that's correct.Therefore, when we solve for s, we get s = x√2. Since s must be between 0 and1, x must be between 0 and1/√2. So the problem says "P is a point on AC different from A and C", but if x can go up to √2, but in reality, x is constrained to 0 ≤ x ≤1/√2 so that s remains ≤1. Hmm. Maybe there's a misunderstanding here.Wait, perhaps the problem allows E to be beyond D? But no, the problem states that E is on CD. So CD is from C(1,1) to D(0,1), so E must lie between (1,1) and (0,1). Therefore, s must be between 0 and1, so x√2 must be between 0 and1. Therefore, x is between0 and1/√2. Therefore, the problem statement might have a typo, or perhaps my reasoning is wrong. Wait, but the problem says P is on AC different from A and C, so x is between0 and√2, but perhaps the case where x >1/√2, E would not be on CD. Maybe there is another position? Wait, maybe my mistake is in the direction of CD.Wait, CD is from C(1,1) to D(0,1). So moving from C to D, the x-coordinate decreases from1 to0, while y remains1. So point E on CD has coordinates (s,1) where s ranges from1 to0. Wait, but usually, when we parametrize a line segment, we take the parameter from0 to1. So if we let E be (1 - t,1) where t ranges from0 to1, then when t=0, E is at C, and when t=1, E is at D. But in my previous reasoning, I let E be (s,1) with s from0 to1, but in reality, s should be from1 to0. Wait, but in coordinate terms, the x-coordinate of CD goes from1 to0 as we move from C to D. So if I take s as the x-coordinate, then s ∈ [0,1] would actually place E from D to C, but perhaps that's just a matter of parameterization. But since we obtained s = x√2, and s must be in [0,1], x must be in [0,1/√2]. Therefore, even though P is on AC (from A(0,0) to C(1,1)), the position of E restricts x to [0,1/√2]. Otherwise, E would be outside CD. So maybe that's a constraint given implicitly in the problem.Therefore, we can proceed with s = x√2, which gives us the x-coordinate of E as s = x√2, so E is at (x√2,1). Therefore, the coordinates of E are (x√2,1). Let me confirm that when x =0, E is at (0,1), which is point D. When x =1/√2, E is at (1,1), which is point C. But the problem states E is on CD, so when x approaches1/√2, E approaches C, and when x approaches0, E approaches D. So this seems consistent.Okay, so now that we have E at (x√2,1), we can move on to finding the area of the pentagon ABPED.The pentagon ABPED has vertices at A(0,0), B(1,0), P(x/√2, x/√2), E(x√2,1), and D(0,1). Let me list the coordinates:A: (0,0)B: (1,0)P: (x/√2, x/√2)E: (x√2,1)D: (0,1)To find the area of this pentagon, one way is to use the shoelace formula. Let's list the coordinates in order and apply the formula.Order of points: A(0,0), B(1,0), P(x/√2, x/√2), E(x√2,1), D(0,1), back to A(0,0).Shoelace formula: For points (x1,y1), (x2,y2), ..., (xn,yn), the area is 1/2 |sum from 1 to n of (xi yi+1 - xi+1 yi)|.So let's compute each term:First pair: A to B: (0,0) to (1,0)Term: x1 y2 - x2 y1 = 0*0 - 1*0 = 0Second pair: B to P: (1,0) to (x/√2, x/√2)Term: x2 y3 - x3 y2 = 1*(x/√2) - (x/√2)*0 = x/√2Third pair: P to E: (x/√2, x/√2) to (x√2,1)Term: x3 y4 - x4 y3 = (x/√2)*1 - (x√2)*(x/√2) = x/√2 - x√2*(x/√2)Simplify: x/√2 - x^2*(√2/√2) = x/√2 - x²Fourth pair: E to D: (x√2,1) to (0,1)Term: x4 y5 - x5 y4 = (x√2)*1 - 0*1 = x√2Fifth pair: D to A: (0,1) to (0,0)Term: x5 y1 - x1 y5 = 0*0 - 0*1 = 0Sum all these terms:0 + x/√2 + (x/√2 - x²) + x√2 + 0 = x/√2 + x/√2 - x² + x√2Combine like terms:( x/√2 + x/√2 ) + x√2 - x² = (2x)/√2 + x√2 - x²Simplify 2x/√2 = x√2:x√2 + x√2 - x² = 2x√2 - x²Then the area is 1/2 |2x√2 - x²|. Since all terms are positive when x is between0 and1/√2, we can drop the absolute value:Area y = (1/2)(2x√2 - x²) = x√2 - (x²)/2Therefore, the relationship between y and x is y = x√2 - (x²)/2.Wait, let me check that again. When we computed the shoelace terms:First term: 0Second term: x/√2Third term: x/√2 - x²Fourth term: x√2Fifth term:0Sum: x/√2 + (x/√2 - x²) + x√2 = x/√2 + x/√2 + x√2 - x² = (2x)/√2 + x√2 - x²But 2x/√2 = x√2, so total is x√2 + x√2 - x² = 2x√2 - x². Then multiply by 1/2: y = (2x√2 - x²)/2 = x√2 - (x²)/2. That seems right.But let me verify with a specific case. For example, when x =0, the pentagon becomes ABDED, but when x=0, P is at A, but the problem states P is different from A and C. So approaching x=0, the area should approach the area of the pentagon ABPED when P approaches A. When x approaches0, E approaches D, so the pentagon becomes almost the quadrilateral ABDED, which is actually the triangle ABD plus the "degenerate" quadrilateral? Wait, when x=0, P coincides with A, so the pentagon becomes A, B, A, E, D, which is not a valid pentagon. Therefore, perhaps we need to check another point.Take x =1/√2, which is the maximum x can be. Then s = x√2 =1/√2 *√2 =1, so E is at (1,1), which is point C. But the pentagon becomes ABPCD, where P is at (1/√2,1/√2). Let's compute its area.Alternatively, compute y = x√2 - (x²)/2. If x=1/√2, then y= (1/√2)(√2) - ( (1/√2)^2 ) /2 =1 - (1/2)/2 =1 -1/4= 3/4. Let's check this.The square has area1. The pentagon ABPED when x=1/√2 would be ABPCD. The area not included is the triangle PBC. Let's compute the area of triangle PBC. Point P is (1/√2,1/√2), B is (1,0), C is (1,1).The area of triangle PBC can be found using coordinates. Coordinates of P(1/√2,1/√2), B(1,0), C(1,1). Using shoelace formula:Area = 1/2 | (1/√2)(0 -1) +1(1 -1/√2) +1(1/√2 -0) |.Calculate each term:First term:1/√2*(-1) = -1/√2Second term:1*(1 -1/√2) =1 -1/√2Third term:1*(1/√2) =1/√2Sum: -1/√2 +1 -1/√2 +1/√2 = -1/√2 +1 -1/√2 +1/√2 =1 -1/√2Therefore, area is 1/2 |1 -1/√2 | = (1 -1/√2)/2.Therefore, the area of the pentagon ABPCD would be area of square (1) minus area of triangle PBC: 1 - (1 -1/√2)/2 =1 -1/2 +1/(2√2)=1/2 +1/(2√2). But according to our formula y = x√2 -x²/2, when x=1/√2, y= (1/√2)(√2) - (1/2)(1/2)=1 -1/4=3/4.But 1/2 +1/(2√2) is approximately 0.5 +0.3535=0.8535, which is not equal to 3/4=0.75. Therefore, there's a discrepancy here. So something's wrong.Wait, this suggests that my earlier calculation is incorrect. Let's check again.Alternatively, maybe the shoelace formula was applied incorrectly. Let me reapply the shoelace formula step by step.List the coordinates in order:A(0,0), B(1,0), P(x/√2, x/√2), E(x√2,1), D(0,1), back to A(0,0).Compute the sum of xi yi+1 - xi+1 yi for each i.1. i=1: x1=0, y1=0; x2=1, y2=0. Term:0*0 -1*0=02. i=2: x2=1, y2=0; x3=x/√2, y3=x/√2. Term:1*(x/√2) - (x/√2)*0 =x/√23. i=3: x3=x/√2, y3=x/√2; x4=x√2, y4=1. Term:(x/√2)*1 -x√2*(x/√2)=x/√2 -x²4. i=4: x4=x√2, y4=1; x5=0, y5=1. Term:x√2*1 -0*1 =x√25. i=5: x5=0, y5=1; x6=0, y6=0. Term:0*0 -0*1=06. i=6: x6=0, y6=0; x1=0, y1=0. Not needed since we already looped back.Total sum:0 +x/√2 +x/√2 -x² +x√2 +0= x/√2 +x/√2 +x√2 -x²= (2x)/√2 +x√2 -x²= x√2 +x√2 -x²=2x√2 -x²Multiply by1/2: (2x√2 -x²)/2= x√2 -x²/2. So that's the same as before.But when x=1/√2, y=√2*(1/√2) - (1/√2)^2 /2=1 - (1/2)/2=1 -1/4=3/4. But the actual area computed by subtracting triangle PBC gives 1 - [(1 -1/√2)/2]= (1 +1/√2)/2≈(1 +0.707)/2≈0.8535, which is not 3/4=0.75. Therefore, inconsistency. Therefore, there must be a mistake in the application of the shoelace formula or in the coordinates of the points.Wait, perhaps the order of the points in the pentagon is incorrect? Let's visualize the pentagon ABPED.Starting at A(0,0), then B(1,0), then P(x/√2, x/√2), then E(x√2,1), then D(0,1). Connecting these points in order.But when x=1/√2, P is (1/√2,1/√2), E is (1,1). So the pentagon is A(0,0), B(1,0), P(1/√2,1/√2), E(1,1), D(0,1). Connecting these points.Wait, when connecting P(1/√2,1/√2) to E(1,1), that's a diagonal line from the center towards C. Then E(1,1) to D(0,1) is the top side. Then back to A. But perhaps this is forming a pentagon, but when we compute the area via shoelace, it's giving 3/4, but the actual area should be different.Alternatively, maybe there's an error in the coordinates of E. Let me check.Earlier, we found that s =x√2, so E is at (x√2,1). But if x=1/√2, then E is at (1,1), which is correct. So E coincides with C. Then the pentagon is ABPCD, which is the quadrilateral ABCDA minus the triangle PBC? Wait, no, ABPCD would be the pentagon going A to B to P to C to D to A. Wait, but according to our coordinates, when x=1/√2, E is at C, so the pentagon is A, B, P, C, D, which is indeed a pentagon. The area should be computed correctly.Wait, using shoelace formula for A(0,0), B(1,0), P(1/√2,1/√2), C(1,1), D(0,1):Coordinates:(0,0), (1,0), (1/√2,1/√2), (1,1), (0,1), back to (0,0)Compute the sum:1. 0*0 -1*0=02.1*(1/√2) - (1/√2)*0=1/√23.(1/√2)*1 -1*(1/√2)=1/√2 -1/√2=04.1*1 -0*1=15.0*0 -0*1=0Sum:0 +1/√2 +0 +1 +0=1 +1/√2Multiply by1/2: (1 +1/√2)/2≈(1 +0.707)/2≈0.8535But according to our formula y= x√2 -x²/2, when x=1/√2, y=3/4≈0.75. This discrepancy shows that there's a mistake in the shoelace application earlier. Wait, why is there a difference?Wait, in the initial problem, the pentagon is ABPED. When x=1/√2, E is at C(1,1), so the pentagon is ABPCD. But according to the shoelace formula, the area is (1 +1/√2)/2≈0.8535, but according to our formula, it's 3/4≈0.75. So my formula is wrong.Therefore, there must be a mistake in the coordinates or in the order of the points.Wait, let me check the coordinates again. When I applied the shoelace formula initially, I took the points in the order A, B, P, E, D. But when E is at C, which is (1,1), then the order should be A(0,0), B(1,0), P(1/√2,1/√2), C(1,1), D(0,1). However, in my initial calculation, for general E, I used E(x√2,1). But when x=1/√2, x√2=1, so E is at (1,1). But in the general case, is the order of the points correct?Wait, the pentagon is ABPED. So after P, we go to E, then to D. But when E is on CD, moving from P to E to D, the path might not be a simple polygon. Let me check with x=1/√2. Then the pentagon is A(0,0), B(1,0), P(1/√2,1/√2), E(1,1), D(0,1). Connecting these points in order, it should form a pentagon without intersecting sides.But when applying the shoelace formula, the area should be calculated correctly. However, there's a discrepancy between the two methods. This suggests that the mistake is in the initial assumption or calculation.Wait, maybe the problem is with the parameterization of point E. Earlier, when we derived s = x√2, but when x=1/√2, s=1, which is correct. So E is at (1,1). Then the shoelace formula using the coordinates A, B, P, E, D gives area y= x√2 -x²/2, which for x=1/√2 is 3/4. But when computed directly, it's (1 +1/√2)/2≈0.8535. Therefore, there's a conflict.This suggests that the error is in the derivation of s. Let me re-examine the earlier steps where we solved for s.We had vectors PB and PE:PB = (1 - x/√2, -x/√2)PE = (s - x/√2, 1 - x/√2)Their dot product is zero:(1 - x/√2)(s - x/√2) + (-x/√2)(1 - x/√2) =0Expanding this:(1)(s) - (1)(x/√2) - (x/√2)(s) + (x²)/2 - (x/√2)(1) + (x²)/2 =0Wait, wait. Let me expand the dot product correctly:PB • PE = (1 - x/√2)(s - x/√2) + (-x/√2)(1 - x/√2)First term: (1)(s) + (1)(-x/√2) + (-x/√2)(s) + (-x/√2)(-x/√2)= s - x/√2 - (x s)/√2 + x²/2Second term: (-x/√2)(1) + (-x/√2)(-x/√2)= -x/√2 + x²/2Therefore, total:s - x/√2 - (x s)/√2 + x²/2 - x/√2 + x²/2 =0Combine like terms:s - 2x/√2 - (x s)/√2 + x² =0Which is:s - x√2 - (x s)/√2 +x²=0Then:s(1 - x/√2) =x√2 -x²Therefore:s= (x√2 -x²)/(1 -x/√2)Factor numerator and denominator:Numerator: x(√2 -x)Denominator: (√2 -x)/√2Therefore, s= [x(√2 -x)] / [ (√2 -x)/√2 ] =x√2Thus, s=x√2. So this seems correct. But when x=1/√2, s=1, which is correct.But then why does the shoelace formula give different results? Let me check the coordinates for x=1/√2:A(0,0), B(1,0), P(1/√2,1/√2), E(1,1), D(0,1)Using shoelace formula:Sum the terms:0*0 -1*0=01*(1/√2) - (1/√2)*0=1/√2(1/√2)*1 -1*(1/√2)=01*1 -0*1=10*0 -0*1=0Sum:0 +1/√2 +0 +1 +0=1 +1/√2Area=1/2*(1 +1/√2)≈0.8535, but according to our formula y= x√2 -x²/2=1 - (1/2)/2=1 -1/4=3/4=0.75This is a contradiction. Therefore, my earlier derivation must be wrong.Wait, what is the correct area? If we compute the pentagon ABPED when x=1/√2, it's supposed to be ABPCD. Let's calculate its area manually.The pentagon consists of the square minus the area of triangle PBC and triangle AED. Wait, no, when E is at C, AED is not a triangle. Wait, maybe another approach.Alternatively, divide the pentagon into simpler shapes. From A to B to P to E to D to A.For x=1/√2, this is A(0,0), B(1,0), P(1/√2,1/√2), E(1,1), D(0,1). This can be divided into:1. The triangle ABP.2. The quadrilateral BPED.But maybe a better way is to use the shoelace formula correctly. Wait, maybe I missed a term.Wait, in my initial application, I listed the coordinates as A, B, P, E, D. But perhaps the order is not correct? Let me confirm the order of the pentagon.Pentagon ABPED: starting at A, going to B, then to P, then to E, then to D, then back to A. So the coordinates should be ordered without crossing.Plotting the points:A(0,0), B(1,0), P(1/√2,1/√2), E(1,1), D(0,1). Connecting these in order, the polygon is non-intersecting, so shoelace should work.But when using shoelace, the area is 1/2*(1 +1/√2). But according to our formula, it's 3/4. Therefore, the mistake must be in the derivation of the formula.Wait, perhaps I made a mistake in the sign when applying the shoelace formula. Let me recompute the shoelace sum carefully.Coordinates:A(0,0), B(1,0), P(x/√2, x/√2), E(x√2,1), D(0,1), back to A(0,0)Compute xi*yi+1 - xi+1*yi for each i:1. i=1: A to B: 0*0 -1*0=02. i=2: B to P:1*(x/√2) - (x/√2)*0=x/√23. i=3: P to E: (x/√2)*1 - (x√2)*(x/√2)=x/√2 -x²4. i=4: E to D: x√2*1 -0*1=x√25. i=5: D to A:0*0 -0*1=06. Total sum:0 +x/√2 +x/√2 -x² +x√2 +0=2x/√2 +x√2 -x²Simplify 2x/√2=x√2, so total isx√2 +x√2 -x²=2x√2 -x². Multiply by1/2:y= x√2 -x²/2.But when x=1/√2, this gives y=1 - (1/2)/2=1 -1/4=3/4, but manual calculation gives (1 +1/√2)/2≈0.8535. The discrepancy is due to different results. Therefore, one of the steps must be wrong.Wait, perhaps the coordinates of E are incorrect. Wait, when we derived s =x√2, but in the problem statement, E is on CD, which is from C(1,1) to D(0,1). So the x-coordinate of E should be between0 and1. But when x=1/√2, s=x√2=1, which is correct. So coordinates of E are (s,1)=(1,1), which is point C. So that's okay.But when we apply the shoelace formula with the points A(0,0), B(1,0), P(1/√2,1/√2), C(1,1), D(0,1), the area is1/2*(1 +1/√2). But according to our formula, it's3/4. So why the difference?Wait, maybe the issue is that when E is at C, the pentagon becomes ABPCD, which includes the triangle BPC. But according to the shoelace formula with the given coordinates, it's computing the area correctly. Therefore, the formula y=x√2 -x²/2 must be incorrect.Alternatively, perhaps the problem is in the definition of the area of the pentagon. If we think about the pentagon ABPED, when E is at C, then it's ABPCD. But according to the area computed via shoelace formula, it's (1 +1/√2)/2≈0.8535, but according to our derived formula, it's3/4≈0.75. These are different. Therefore, there's a mistake in the derivation.Wait, another way to compute the area is to add up the areas of the components. For the pentagon ABPED, it's the area of square ABCD minus the areas of triangle PBC and triangle EDA.Wait, but when E is at C, triangle EDA becomes triangle CDA, which is half the square, but that might not be the case.Alternatively, the pentagon ABPED can be divided into three parts: triangle ABP, quadrilateral BPED, and triangle EDA? Maybe not. Let's try.Alternatively, perhaps the pentagon ABPED can be split into triangle ABP, quadrilateral BPED, and triangle EDA. But need to visualize.Alternatively, compute the area step by step.From point A(0,0) to B(1,0) to P(x/√2, x/√2) to E(x√2,1) to D(0,1) to A(0,0).The area can be calculated as the sum of the trapezoid ABPE and the triangle EDA. Wait, not sure.Alternatively, use the shoelace formula with correct coordinates. Wait, when x=1/√2, coordinates are A(0,0), B(1,0), P(1/√2,1/√2), E(1,1), D(0,1). Shoelace gives:Sum of terms:0*0 -1*0=01*(1/√2) - (1/√2)*0=1/√2(1/√2)*1 -1*(1/√2)=01*1 -0*1=10*0 -0*1=0Total sum:0 +1/√2 +0 +1 +0=1 +1/√2Area=1/2*(1 +1/√2). But according to our formula, y=3/4.This inconsistency suggests that my initial derivation is incorrect. Therefore, there must be a mistake in the setup.Wait, let's consider another approach. Let's parametrize the problem using coordinates where the square is from (0,0) to (1,1). Let's assume A(0,0), B(1,0), C(1,1), D(0,1). Diagonal AC has equation y=x.AP =x. The distance from A to P along AC is x, so coordinates of P are (x/√2, x/√2).Point E is on CD, which is from (1,1) to (0,1). So E has coordinates (s,1), where 0 ≤s ≤1.PE is perpendicular to PB.Vector PB is from P to B: (1 -x/√2, -x/√2)Vector PE is from P to E: (s -x/√2,1 -x/√2)Their dot product is zero:(1 -x/√2)(s -x/√2) + (-x/√2)(1 -x/√2)=0Earlier, we solved this to find s=x√2.But when x=1/√2, s=1, which is correct.But according to shoelace formula, the area when x=1/√2 is (1 +1/√2)/2, but according to y=x√2 -x²/2, it's 3/4. Therefore, there's a contradiction. Therefore, there must be a mistake in the derivation.Wait, maybe I made a mistake in interpreting the direction of the vectors. The vectors PB and PE are vectors from P to B and P to E, but in the problem statement, PE is perpendicular to PB. So the vectors are PE and PB, which are from P to E and P to B, respectively. Therefore, their dot product is zero.Yes, that's what was done. So that part is correct.Alternatively, maybe the error is in the shoelace formula application. When I applied shoelace, I assumed the order A, B, P, E, D. But perhaps this order causes overlapping or negative areas?Wait, let's plot the points for x=1/2.Take x=1/2. Then P is at (1/(2√2),1/(2√2))≈(0.3535,0.3535). E is at s= x√2=1/2*1.414≈0.707, so E is at (0.707,1).The pentagon A(0,0), B(1,0), P(0.3535,0.3535), E(0.707,1), D(0,1). Connecting these points, the area calculated via shoelace formula should be:Compute the terms:0*0 -1*0=01*0.3535 -0.3535*0=0.35350.3535*1 -0.707*0.3535≈0.3535 -0.25≈0.10350.707*1 -0*1≈0.7070*0 -0*1=0Sum≈0 +0.3535 +0.1035 +0.707 +0≈1.164Area≈1/2*1.164≈0.582According to our formula, y= x√2 -x²/2= (1/2)*1.414 - (1/4)/2≈0.707 -0.125=0.582. So in this case, the formula matches.But for x=1/√2, the formula gives y=3/4=0.75, while the manual calculation gives≈0.8535. Therefore, the discrepancy is only at x=1/√2. What's happening?Wait, when x=1/√2≈0.707, and s= x√2=1, which is point C(1,1). So the pentagon is A(0,0), B(1,0), P(0.707,0.707), E(1,1), D(0,1). Applying shoelace formula:Coordinates:A(0,0), B(1,0), P(0.707,0.707), E(1,1), D(0,1), back to A(0,0)Compute terms:1. 0*0 -1*0=02.1*0.707 -0.707*0=0.7073.0.707*1 -1*0.707=04.1*1 -0*1=15.0*0 -0*1=0Sum=0 +0.707 +0 +1 +0=1.707Area=1/2*1.707≈0.8535, which is (1 +1/√2)/2≈(1 +0.707)/2≈0.8535. But according to our formula, y= x√2 -x²/2=1 - ( (√2)^2/(2*2) )=1 - (2)/4=1 -0.5=0.5. Wait, no. Wait, x=1/√2≈0.707.Wait, x=1/√2, then x√2=1, and x²=1/2. Therefore, y=1 - (1/2)/2=1 -1/4=3/4=0.75. But manual calculation gives0.8535. Therefore, there's a mistake.Wait, the issue arises because when E is at C(1,1), the pentagon ABPED is actually ABPCD. However, our formula computes this area as3/4, but manual computation gives (1 +1/√2)/2≈0.8535. Therefore, there's a contradiction, suggesting the formula is incorrect.But for x=1/2, the formula works. So why the discrepancy at x=1/√2?Wait, perhaps when E coincides with C, the pentagon ABPED is not the same as ABPCD? Wait, ED would be from E=C to D, which is CD. So ABPED would be ABPCD. But in that case, the area should be consistent.Alternatively, perhaps the shoelace formula is being applied incorrectly when the polygon folds over itself? But in this case, the polygon is simple.Wait, maybe the formula y= x√2 -x²/2 is correct, but the manual calculation is wrong. Let's compute y= x√2 -x²/2 for x=1/√2:y= (1/√2)*√2 - (1/√2)^2 /2=1 - (1/2)/2=1 -1/4=3/4=0.75.But when calculated via shoelace, it's0.8535. Therefore, the formula is wrong.This suggests that the earlier derivation has an error. Let's re-examine the shoelace formula application.The coordinates are A(0,0), B(1,0), P(x/√2, x/√2), E(x√2,1), D(0,1). Applying shoelace formula:Sum terms:1. 0*0 -1*0=02.1*(x/√2) - (x/√2)*0=x/√23.(x/√2)*1 -x√2*(x/√2)=x/√2 -x²4.x√2*1 -0*1=x√25.0*0 -0*1=0Sum:0 +x/√2 +x/√2 -x² +x√2 +0=2x/√2 +x√2 -x²= x√2 +x√2 -x²=2x√2 -x²Then area= (2x√2 -x²)/2= x√2 -x²/2.But when x=1/√2, this is1 - (1/2)/2=1 -1/4=3/4. But manual calculation using coordinates gives (1 +1/√2)/2≈0.8535. There's a discrepancy. Therefore, the formula must be incorrect.But why? What is the error?Wait, perhaps the order of the points is not correct. In the shoelace formula, the order of the points must be either clockwise or counter-clockwise without crossing. Let me check the order:A(0,0), B(1,0), P(x/√2, x/√2), E(x√2,1), D(0,1).Plotting these points:A is at origin.B is at (1,0).P is somewhere along the diagonal AC.E is on CD.D is at (0,1).Connecting these in order: A to B to P to E to D to A. This should form a pentagon. But when E is at C, this pentagon includes the edge PE which is from P to C, then to D. However, when applying the shoelace formula, the area should be calculated correctly. But when x=1/√2, the shoelace formula gives a different result.Alternatively, maybe the problem is that when E is at C, the pentagon ABPED becomes a quadrilateral ABPCD, which has four vertices, but we are listing five points. Wait, no, ABPCD is a pentagon with vertices A, B, P, C, D.But when E coincides with C, the pentagon ABPED becomes ABPCD. The coordinates are still five points: A, B, P, C, D. Therefore, the shoelace formula should work.But according to our formula, the area is3/4, but manual calculation gives (1 +1/√2)/2≈0.8535. So which is correct?Let's compute the area using another method. For x=1/√2, the pentagon ABPCD can be divided into triangle ABP and quadrilateral BPCD.Area of triangle ABP: base AB=1, height is the y-coordinate of P, which is1/√2≈0.707. So area=1/2 *1 *1/√2≈0.3535.Area of quadrilateral BPCD: Points B(1,0), P(1/√2,1/√2), C(1,1), D(0,1). This can be divided into triangle BPC and trapezoid PCDA?Wait, point D is (0,1). So quadrilateral BPCD is B(1,0), P(1/√2,1/√2), C(1,1), D(0,1). To find its area, use shoelace formula:Coordinates: B(1,0), P(1/√2,1/√2), C(1,1), D(0,1), back to B(1,0)Sum:1*(1/√2) - (1/√2)*0=1/√2(1/√2)*1 -1*(1/√2)=01*1 -0*1=10*0 -1*1=-1Sum:1/√2 +0 +1 -1=1/√2≈0.707Area=1/2*0.707≈0.3535But adding to the area of triangle ABP (0.3535), total area≈0.707, which is3/4≈0.75. Wait, no. Wait, total area would be0.3535 +0.3535=0.707, but according to manual shoelace formula, the area of the pentagon is≈0.8535. Therefore, there's confusion.This indicates that something is fundamentally wrong with the approach. Maybe the initial assumption that PE is perpendicular to PB leads to E being on CD only when x ≤1/√2, and for x >1/√2, E would be outside CD. Therefore, the problem implicitly assumes x ∈[0,1/√2]. However, even within this interval, the area formula derived via shoelace doesn't match manual calculations for certain points.This suggests that the error is in the initial formulation. Alternatively, perhaps the problem is designed in such a way that the area is indeed y =x√2 -x²/2, and the discrepancy arises due to miscalculating manual examples.Alternatively, accept the derived formula y =x√2 -x²/2 as correct and attribute the discrepancy at x=1/√2 to an error in manual calculation. But given that for x=1/2, the formula matches, but for x=1/√2, it doesn't, there's inconsistency.Alternatively, consider that when E is at C, the pentagon ABPED is not convex and the shoelace formula may not apply correctly. However, in this case, the polygon remains non-intersecting, so shoelace should work.Alternatively, there's an error in the problem setup. Maybe the correct answer is y= x√2 -x²/2, and the discrepancy is due to my miscalculation. Let me compute the shoelace area for x=1/√2 again.Coordinates:A(0,0), B(1,0), P(1/√2,1/√2), E(1,1), D(0,1)Compute terms:1. 0*0 -1*0=02.1*(1/√2) - (1/√2)*0=1/√23.(1/√2)*1 -1*(1/√2)=04.1*1 -0*1=15.0*0 -0*1=0Sum=0 +1/√2 +0 +1 +0=1 +1/√2Area=1/2*(1 +1/√2)= (√2 +1)/(2√2)= (√2 +1)/(2√2)*√2/√2= ( (√2 +1)√2 ) /4= (2 +√2)/4≈(2 +1.414)/4≈3.414/4≈0.8535But according to y= x√2 -x²/2=1 -1/4=3/4=0.75.This suggests that the formula is incorrect. Therefore, there must be a mistake in the derivation.Alternatively, perhaps the area of the pentagon is not calculated correctly using the given coordinates. Let me consider another approach.The pentagon ABPED can be divided into three parts: triangle ABP, quadrilateral BPEB, and triangle EDA. Wait, not sure.Alternatively, integrate the area under certain curves.Alternatively, parametrize the problem differently.Let me try another approach without coordinates. Since the square has side 1, and AP =x along the diagonal AC. Since AC has length √2, x is between0 and√2.PE is perpendicular to PB. Let me consider the positions of points P and E.Since PE ⊥ PB, the product of their slopes is -1.Slope of PB: (B - P) coordinates: B is (1,0), P is (x/√2, x/√2). So slope is (0 - x/√2)/(1 - x/√2)= (-x/√2)/(1 -x/√2)Slope of PE: (E - P) coordinates: E is (s,1), P is (x/√2, x/√2). So slope is (1 -x/√2)/(s -x/√2)Their product must be -1:[ (-x/√2)/(1 -x/√2) ] * [ (1 -x/√2)/(s -x/√2) ] = -1Simplify:[ (-x/√2)(1 -x/√2) ] / [ (1 -x/√2)(s -x/√2) ] = -1The (1 -x/√2) terms cancel out:(-x/√2)/(s -x/√2) = -1Multiply both sides by (s -x/√2):-x/√2 = - (s -x/√2)Multiply both sides by -1:x/√2 = s -x/√2Therefore:s = 2x/√2 =x√2Which is the same result as before. So s=x√2 is correct.Therefore, the coordinates are correct, and the shoelace formula should give the correct area, but there's a discrepancy when x=1/√2.This suggests that the formula y=x√2 -x²/2 is correct, and the manual calculation for x=1/√2 is wrong. But why?Wait, when x=1/√2, y=3/4. The square has area1, so the area of the pentagon is3/4, meaning the remaining area is1/4. The remaining area would be the triangle PBC and the triangle EDA. But when E is at C, EDA is triangle CDA, which has area1/2. But this contradicts.Wait, no. If the pentagon area is3/4, the remaining area would be1 -3/4=1/4. But how?Wait, when E is at C, the remaining area is the triangle PBC. So area of PBC should be1/4. Let's compute area of PBC when x=1/√2.Point P is(1/√2,1/√2), B(1,0), C(1,1).Using shoelace formula for triangle PBC:Coordinates: P(1/√2,1/√2), B(1,0), C(1,1)Area=1/2| (1/√2)(0 -1) +1(1 -1/√2) +1(1/√2 -0) |=1/2| (-1/√2) +1 -1/√2 +1/√2 |=1/2| (-1/√2) +1 -1/√2 +1/√2 |=1/2|1 -1/√2|= (1 -1/√2)/2≈(1 -0.707)/2≈0.146But1 -3/4=1/4=0.25, which does not match. Therefore, the remaining area is not just triangle PBC. Therefore, my assumption is wrong.Wait, if pentagon ABPED has area3/4, then the remaining area would include the quadrilateral PBC and the triangle EDA. But when E is at C, EDA is triangle CDA, which has area0.5. But 0.5 +0.146=0.646, which added to0.75 exceeds1. So this approach is flawed.Therefore, it's clear that there's a mistake somewhere, but I can't pinpoint it. Given the time I've spent and the steps I've verified multiple times, I think the derived formula y =x√2 -x²/2 is correct, and the manual calculation discrepancy must be due to an error in the manual area calculation for specific cases. Perhaps the shoelace formula was applied incorrectly due to incorrect coordinate ordering or arithmetic mistake.Given that for x=1/2, the formula works, and the derivation seems correct, I will proceed with the answer y= x√2 - (x²)/2.Final AnswerThe relationship between ( y ) and ( x ) is (boxed{y = sqrt{2}x - dfrac{x^2}{2}}).</think>Given a square (ABCD) with side length 1, point (P) is on the diagonal (AC) such that (AP = x). Point (E) is on (CD) such that (PE perp PB). We need to find the relationship between the area (y) of the pentagon (ABPED) and (x).1. Coordinates of Points: - (A(0,0)), (B(1,0)), (C(1,1)), (D(0,1)). - Point (P) on (AC) has coordinates (left(frac{x}{sqrt{2}}, frac{x}{sqrt{2}}right)) since (AP = x).2. Finding Coordinates of (E): - Point (E) is on (CD), so its coordinates are ((s, 1)). - Vectors (PB) and (PE) must be perpendicular. Using the dot product condition: [ left(1 - frac{x}{sqrt{2}}right)left(s - frac{x}{sqrt{2}}right) + left(-frac{x}{sqrt{2}}right)left(1 - frac{x}{sqrt{2}}right) = 0 ] - Solving this, we find (s = xsqrt{2}). Thus, (E) has coordinates ((xsqrt{2}, 1)).3. Area Calculation Using Shoelace Formula: - Coordinates of the pentagon (ABPED): (A(0,0)), (B(1,0)), (Pleft(frac{x}{sqrt{2}}, frac{x}{sqrt{2}}right)), (E(xsqrt{2}, 1)), (D(0,1)). - Applying the shoelace formula: [ text{Area} = frac{1}{2} left| 0 + frac{x}{sqrt{2}} + left(frac{x}{sqrt{2}} - x^2right) + xsqrt{2} + 0 right| ] - Simplifying the expression: [ text{Area} = frac{1}{2} left( 2xsqrt{2} - x^2 right) = xsqrt{2} - frac{x^2}{2} ]Thus, the relationship between (y) and (x) is (boxed{y = sqrt{2}x - dfrac{x^2}{2}}).,<think>Okay, so I have this geometry problem here. Let me read it again carefully. We have a square ABCD with side length 1. Point P is on the diagonal AC, but not at A or C. Then there's a point E on CD such that PE is perpendicular to PB. We need to find the relationship between y, which is the area of the pentagon ABPED, and x, which is the length AP. First, I need to visualize the square. Let me sketch it mentally. Let's say square ABCD has vertices A at the bottom-left corner, B at the bottom-right, C at the top-right, and D at the top-left. The diagonal AC goes from A to C. So point P is somewhere along this diagonal, and AP is x. Since the square has side length 1, the diagonal AC should have length √2, right? So AP = x, which means PC = √2 - x. Now, point E is on CD. CD is the top side from C to D. So E is somewhere along that edge. The condition given is that PE is perpendicular to PB. So PE and PB are perpendicular lines. Our goal is to find the area of pentagon ABPED in terms of x. Let me think. To find the area of the pentagon, maybe I can subtract the areas not included in the pentagon from the total area of the square. The square has area 1, so if I can find the area of triangle BPC or something else that's not part of the pentagon, that might help. Alternatively, maybe I can break the pentagon into simpler shapes whose areas I can compute and then add them up.But first, perhaps I need to find the coordinates of points P and E. Using coordinate geometry might make this problem easier. Let's assign coordinates to the square. Let me place point A at (0,0), so B would be (1,0), C at (1,1), and D at (0,1). Then the diagonal AC goes from (0,0) to (1,1). Point P is on AC with AP = x. Since AC has length √2, but in coordinates, moving from (0,0) to (1,1), each step along the diagonal increases both x and y coordinates by 1/√2 per unit length. But maybe it's easier to parameterize point P. Since AP = x, but in the coordinate system, the distance from A to P is x. The coordinates of P can be found by moving x units along AC. However, the coordinates of AC are (t, t) for t from 0 to 1. Wait, but the distance from A to (t, t) is t√2. So if AP = x, then t√2 = x, so t = x / √2. Therefore, coordinates of P are (x/√2, x/√2). Hmm, but maybe this is overcomplicating. Wait, since in coordinate terms, moving from A(0,0) to C(1,1), the parametric equations are x = t, y = t for t from 0 to 1. The distance from A to any point (t, t) on AC is t√2. So AP = x implies t√2 = x, so t = x / √2. Hence, coordinates of P are (x/√2, x/√2). But maybe it's simpler to use coordinate system where the square is from (0,0) to (1,1). Let me confirm. If I consider AP = x, then since AC is the diagonal, which in coordinate terms is length √2, but if x is the actual distance, not the coordinate distance, then coordinates of P would be (x/√2, x/√2). That seems correct.Alternatively, if we use coordinates where each side is length 1, then the coordinates of P can be written as (t, t) where t ranges from 0 to 1. Then the distance AP is t√2, so x = t√2, hence t = x / √2. So yes, coordinates of P are (x/√2, x/√2). Now, point E is on CD. CD is from (1,1) to (0,1). So coordinates of E can be written as (s, 1) where s ranges from 0 to 1. We need to find s such that PE is perpendicular to PB. Let me get coordinates for points B, P, E. Point B is (1,0). Point P is (x/√2, x/√2). Point E is (s, 1). First, let's find the vectors PE and PB. Vector PE is E - P = (s - x/√2, 1 - x/√2). Vector PB is B - P = (1 - x/√2, 0 - x/√2). Since PE is perpendicular to PB, their dot product should be zero. So:PE ⋅ PB = 0So:(s - x/√2)(1 - x/√2) + (1 - x/√2)( - x/√2 ) = 0Let me compute this step by step.First term: (s - x/√2)(1 - x/√2)Second term: (1 - x/√2)( -x/√2 )Wait, actually, let's recheck the vectors:PE is from P to E: (s - x/√2, 1 - x/√2)PB is from P to B: (1 - x/√2, -x/√2)So the dot product is:(s - x/√2)(1 - x/√2) + (1 - x/√2)( -x/√2 ) = 0Factor out (1 - x/√2):(1 - x/√2)[ (s - x/√2) + (-x/√2) ] = 0But (1 - x/√2) can't be zero unless x/√2 = 1, which would mean x = √2, but since P is not at C (which is at distance √2 from A), x must be less than √2. However, x is given as AP, which in the problem statement is different from A and C, so x is between 0 and √2. But given that the square has side length 1, the diagonal AC is √2, so x is between 0 and √2. But we need to find the relationship between y and x where x is in (0, √2). However, perhaps the answer is expected in terms of x within 0 to 1? Wait, no, AP is the distance from A to P along the diagonal, so x is from 0 to √2. But maybe in the problem, the coordinates are set up differently.Wait, hold on. Maybe the problem is using AP = x as a length along the diagonal, but in the square with side length 1. But in that case, the diagonal is length √2, so x can be up to √2. However, the answer is supposed to be a relationship between y and x. Maybe we can express it in terms of x without necessarily rationalizing the coordinate system.Alternatively, perhaps the problem is in a coordinate system where the square has side length 1, but AP is given as x, which is a length from 0 to √2. However, when dealing with coordinates, it's often easier to parameterize P as (t, t) where t is from 0 to 1, and then express x as t√2, so t = x / √2. So maybe we can use t as a parameter and express everything in terms of t, then substitute t = x / √2 at the end.But perhaps let me proceed step by step.So, from the dot product equation:(s - x/√2)(1 - x/√2) + (1 - x/√2)(-x/√2) = 0Factor out (1 - x/√2):(1 - x/√2)[ (s - x/√2) - x/√2 ] = 0So, (1 - x/√2)(s - 2x/√2) = 0Since 1 - x/√2 ≠ 0 (because x ≠ √2, as P is not at C), then:s - 2x/√2 = 0 => s = 2x/√2 = x√2Wait, √2/√2 is 1. Wait, 2x / √2 is (2/√2)x = √2 x. So s = √2 x. But s is the x-coordinate of point E on CD, which is from (1,1) to (0,1). Wait, in our coordinate system, CD goes from (1,1) to (0,1). Wait, no. If the square is A(0,0), B(1,0), C(1,1), D(0,1), then CD is from C(1,1) to D(0,1). So CD is the top side from right to left. Therefore, the coordinates of E are (s,1), where s ranges from 0 to 1. But according to the above, s = √2 x. But x is the distance from A to P along AC, which is up to √2. So if x can be up to √2, then s = √2 x could be greater than 1, which is impossible because E is on CD which has s between 0 and 1. Therefore, this suggests that there's a mistake in the calculation.Wait, this is a problem. So maybe my coordinate system assumption is wrong. Let me check again.Wait, in the coordinate system, AP is a distance x along the diagonal AC, which has total length √2. Therefore, x can vary from 0 to √2, but E is on CD, which in coordinates is from (1,1) to (0,1), so s must be between 0 and 1. Therefore, s = √2 x must satisfy 0 ≤ √2 x ≤ 1, which would mean x ≤ 1/√2. But the problem states that P is on AC different from A and C, so x can be from 0 to √2, but in reality, if E is on CD, then √2 x must be ≤1, so x ≤1/√2. Hmm, this is conflicting with the problem statement. Maybe there's an error in my calculation.Let me redo the calculation step by step.Given point P is (x/√2, x/√2). Point E is (s,1). Then vector PE is (s - x/√2, 1 - x/√2). Vector PB is (1 - x/√2, -x/√2). Their dot product must be zero:(s - x/√2)(1 - x/√2) + (1 - x/√2)(-x/√2) = 0Let me factor out (1 - x/√2):(1 - x/√2)[(s - x/√2) + (-x/√2)] = 0So, inside the brackets:s - x/√2 - x/√2 = s - 2x/√2 = s - (2x)/√2Which is s - x√2Thus, (1 - x/√2)(s - x√2) = 0Since 1 - x/√2 ≠ 0 (as x ≠ √2), then s - x√2 = 0 => s = x√2Therefore, s = x√2. But since E is on CD, s must be between 0 and 1, so x√2 ≤1 => x ≤ 1/√2. But the problem states that P is different from A and C, so x is in (0, √2). However, in our calculation, s must be ≤1, so x must be ≤1/√2. Therefore, the problem might have a restriction on x, or maybe I messed up the coordinate system.Wait, maybe the problem is set in a different coordinate system. Let me check again the positions of the square. If the square is labeled A, B, C, D in order, forming a clockwise square, then depending on the starting point. If A is at (0,0), B at (1,0), C at (1,1), D at (0,1), then CD is the top edge from (1,1) to (0,1). But maybe the problem defines the square differently. Alternatively, perhaps I have misassigned the coordinates. Let me confirm.Alternatively, maybe the square is rotated. If the square is labeled such that diagonal AC is from A(0,0) to C(1,1), then yes, CD is from C(1,1) to D(0,1). So E is somewhere on CD, which is the top edge. So E has coordinates (s,1), with s between 0 and1. Therefore, s = x√2 must be ≤1, so x ≤1/√2. But the problem states that P is different from A and C, so x can be in (0, √2). But if x >1/√2, then s = x√2 >1, which would place E outside of CD, which contradicts E being on CD. Therefore, perhaps there is a mistake in my calculation.Wait, perhaps I made an error in the direction of the vectors. Let me re-express vectors PE and PB.Point P is (x/√2, x/√2). Point E is (s,1). Therefore, vector PE is E - P = (s - x/√2, 1 - x/√2). Vector PB is B - P = (1 - x/√2, 0 - x/√2) = (1 - x/√2, -x/√2). The dot product is:(s - x/√2)(1 - x/√2) + (1 - x/√2)(-x/√2) = 0Wait, hold on. The second component of PE is (1 - x/√2), and the second component of PB is (-x/√2). Therefore, the dot product is:First component: (s - x/√2)(1 - x/√2)Second component: (1 - x/√2)(-x/√2)So when you add them together, it's:(s - x/√2)(1 - x/√2) + (1 - x/√2)(-x/√2) = 0Factor out (1 - x/√2):(1 - x/√2)[(s - x/√2) + (-x/√2)] = 0So inside the bracket: s - x/√2 -x/√2 = s - 2x/√2 = s - x√2Therefore, s = x√2.But this still gives s = x√2. So unless x is ≤1/√2, s would exceed 1. Therefore, perhaps the problem has a constraint that x ≤1/√2? Or maybe there's an error in the problem statement. Alternatively, maybe I need to parameterize differently.Wait, maybe instead of parameterizing P by the distance AP = x, which gives coordinates (x/√2, x/√2), perhaps the problem uses a different parameterization. Maybe in the problem, x is the length along AC from A, but the coordinate system is normalized such that the square has side length 1, so the diagonal is √2. But if AP = x, then coordinates of P are (x/√2, x/√2). Alternatively, maybe the problem is using a coordinate system where the square has side length 1, but the diagonal AC is considered as a line from (0,0) to (1,1), so that moving along AC, the coordinates are (t, t) for t from 0 to 1. Then AP, the distance along AC, would be t√2. So if x = t√2, then t = x / √2. Therefore, coordinates of P are (x / √2, x / √2). But in that case, when we derived s = x√2, that's s = t√2 * √2 = t*2. Wait, no. If x = t√2, then s = x√2 = t*2. But t is from 0 to1, so s would be from 0 to2, which can't be. Wait, no. Wait, if x is the distance from A to P along AC, then x = t√2 where t is the coordinate parameter from 0 to1. Then s = x√2 = t*2. Therefore, when t ranges from 0 to 0.5, s ranges from 0 to1. But when t >0.5, s exceeds 1. Therefore, for E to be on CD, which requires s ≤1, t must be ≤0.5, so x must be ≤√2 / 2 ≈0.707. Hence, the problem might only consider x in (0, √2/2). Otherwise, E would not lie on CD.But the problem states that P is on AC different from A and C. It doesn't specify any other constraints, so perhaps in the problem, x is allowed up to √2/2? But the problem statement doesn't mention that. Hmm.Alternatively, maybe my approach is wrong. Maybe I need to use coordinate geometry but parameterize differently. Let's try parameterizing P by a parameter t, where t is the x-coordinate (and y-coordinate since it's on AC). So P is (t, t) where t is between 0 and1. Then AP, the distance from A to P, would be t√2. So x = t√2, hence t = x / √2. Then E is on CD, so coordinates of E are (s,1). Then PE is perpendicular to PB. Let's redo the calculation with this parameterization.Point P: (t, t)Point B: (1,0)Point E: (s,1)Vector PE: (s - t, 1 - t)Vector PB: (1 - t, -t)Their dot product is:(s - t)(1 - t) + (1 - t)(-t) = 0Factor out (1 - t):(1 - t)[(s - t) - t] = 0So (1 - t)(s - 2t) = 0Since 1 - t ≠ 0 (because t ≠1, as P is not at C), then:s - 2t =0 => s =2tTherefore, s =2t. But s must be between 0 and1, so t must be between0 and0.5. Therefore, t ∈ (0,0.5), hence x = t√2 must be in (0, √2/2). Therefore, if we parameterize P by t, then s =2t. Thus, coordinates of E are (2t,1). So now, to find the area of pentagon ABPED. Let's list the coordinates of the pentagon's vertices:A(0,0), B(1,0), P(t, t), E(2t,1), D(0,1). Connecting these in order: A to B to P to E to D to A.To find the area, perhaps we can use the shoelace formula. Let's list the coordinates:A: (0,0)B: (1,0)P: (t, t)E: (2t,1)D: (0,1)Back to A: (0,0)Using shoelace formula:Area = 1/2 |sum over i (x_i y_{i+1} - x_{i+1} y_i)|Compute each term:From A to B: x_i=0, y_i=0; x_{i+1}=1, y_{i+1}=0. Term: 0*0 -1*0 =0From B to P: x_i=1, y_i=0; x_{i+1}=t, y_{i+1}=t. Term:1*t - t*0 = tFrom P to E: x_i=t, y_i=t; x_{i+1}=2t, y_{i+1}=1. Term: t*1 -2t*t = t -2t²From E to D: x_i=2t, y_i=1; x_{i+1}=0, y_{i+1}=1. Term:2t*1 -0*1 =2tFrom D to A: x_i=0, y_i=1; x_{i+1}=0, y_{i+1}=0. Term:0*0 -0*1 =0Sum all terms:0 + t + (t -2t²) +2t +0 = t + t -2t² +2t =4t -2t²Hence, Area =1/2 |4t -2t²| =1/2 (4t -2t²) =2t -t²Since t is between0 and0.5, 4t -2t² is positive.But remember, we need the area in terms of x. Since x = AP = t√2, so t =x /√2.Substitute t into the area formula:Area =2*(x /√2) - (x /√2)² = (2x)/√2 -x² /2 = √2 x - (x²)/2Therefore, the area y =√2 x - (x²)/2.But let's check the shoelace calculation again to make sure.Coordinates:A(0,0); B(1,0); P(t,t); E(2t,1); D(0,1); A(0,0)Shoelace terms:Sum1 (x_i y_{i+1}):0*0 +1*t +t*1 +2t*1 +0*0 =0 + t + t +2t +0=4tSum2 (y_i x_{i+1}):0*1 +0*t +t*2t +1*0 +1*0 =0 +0 +2t² +0 +0=2t²Area =1/2 |Sum1 - Sum2| =1/2 |4t -2t²| =2t -t²Yes, that's correct. Then substitute t =x /√2:y =2*(x /√2) - (x /√2)^2 = (2x)/√2 -x² /2 = √2 x - x² /2Therefore, the relationship is y =√2 x - (1/2)x². But let me check with an example. Suppose t=0.5, then x =0.5√2≈0.707. Then s=1, which is point D? Wait, s=2t=1, so E would be at (1,1), which is point C. But point E is supposed to be on CD. If s=1, then E is at (1,1), which is point C. But the problem states that E is on CD, different from C? Wait, the problem didn't specify E is different from C or D, just that E is on CD. So when t=0.5, E is at C. But in the problem statement, P is different from A and C. So if t=0.5, x=√2/2≈0.707, E is at C. Then the pentagon ABPED would be ABCDED? Wait, no. Let's see, when E is at C, the pentagon would be A, B, P, E=C, D. So A to B to P to C to D to A. That would actually be a hexagon, but since C and D are consecutive, maybe it's considered a pentagon. Hmm. But in that case, the area should be the area of the square minus the area of triangle PCD. Wait, but according to our formula, when t=0.5, area y=√2*(√2/2) - ( (√2/2)^2 )/2= (2/2) - ( (2/4)/2 )=1 - (0.5)/2=1 -0.25=0.75. The area of the square is1, so the pentagon area would be0.75. Alternatively, if E is at C, then the pentagon is ABPCD, which is the square minus triangle APD. APD is a right triangle with legs 0.5 and0.5, area 0.25. So 1 -0.25=0.75. Which matches. So the formula works here.Another test case: when t approaches0, x approaches0. Then y approaches0. But if P approaches A, then the pentagon ABPED becomes ABBED, but since P approaches A, E approaches D (s=2t approaches0). Wait, no. If t approaches0, s=2t approaches0, so E approaches (0,1). So the pentagon would be A, B, A, E approaching D, D. Hmm, not sure. But according to the formula, y=√2*0 -0=0, which might make sense as the area collapses.Alternatively, take t=0.25, x=0.25√2≈0.3535. Then s=0.5. So E is at (0.5,1). Then pentagon is A(0,0), B(1,0), P(0.25,0.25), E(0.5,1), D(0,1). Let's compute its area.Using shoelace formula:Coordinates:A(0,0); B(1,0); P(0.25,0.25); E(0.5,1); D(0,1); A(0,0)Sum1:0*0 +1*0.25 +0.25*1 +0.5*1 +0*0 =0 +0.25 +0.25 +0.5 +0=1Sum2:0*1 +0*0.25 +0.25*0.5 +1*0 +1*0=0 +0 +0.125 +0 +0=0.125Area=1/2 |1 -0.125|=1/2*0.875=0.4375Using our formula: y=√2*(0.25√2) - (0.25√2)^2 /2= (√2*0.25√2)=0.25*2=0.5. Then subtract ( (0.25√2)^2 )/2= (0.0625*2)/2=0.0625. So y=0.5 -0.0625=0.4375. Which matches. So the formula works here.Therefore, the relationship is y=√2 x - (x²)/2.But let me check if there's a way to write this without the radical. Since x is AP, which is along the diagonal, perhaps they expect the answer in terms of x, but maybe expressed differently. Alternatively, maybe rationalizing or using substitution.Alternatively, since in coordinate terms, t = x /√2, and area is 2t - t², so substituting t =x /√2:y=2*(x /√2) - (x /√2)^2 = (2x)/√2 -x² /2= √2 x - (x²)/2Therefore, the relationship is y=√2 x - (1/2)x². So this is a quadratic function in x.But let me see if this is the answer. The problem states "the relationship between y and x", so likely they want an equation in terms of y and x. So the answer is y=√2 x - (1/2)x². But let me verify once again. Alternatively, maybe the problem uses coordinate axes where the diagonal AC is the x-axis. But that complicates things. Alternatively, using coordinate system as we did is standard.Alternatively, perhaps there is another approach to compute the area. Let's see.The pentagon ABPED. Let's break it down into parts.First, the area of square ABCD is1. If we subtract the area not included in the pentagon, which is the area of triangle PEC and triangle BPC? Wait, maybe not. Let's see.Alternatively, the pentagon can be divided into three parts: triangle ABP, quadrilateral BPED, and triangle EDA? Not sure. Alternatively, trapezoid ABED and triangle BPE?Alternatively, think of the pentagon as the combination of triangle ABP, trapezoid BPE D, and triangle EDA?Wait, perhaps using coordinates and integrating, but shoelace formula already gave us the answer. So I think the answer is correct.Thus, the relationship is y=√2 x - (1/2)x².But let me check units. If x is a length, then y is an area. The terms √2 x has units of length, but we need area. Wait, no. Wait, in our calculation, when we substituted t = x /√2 into the area formula y=2t -t², which is in terms of t (unitless if coordinates are unitless). But in the problem, x is a length, so maybe there is confusion here.Wait, no. Wait, in our coordinate system, the square has side length1, so coordinates are from0 to1. Therefore, t is a coordinate between0 and0.5, and x=AP is the distance, which is t√2. Therefore, x is in terms of the unit length. Therefore, when we express y=2t -t², and substitute t=x /√2, we get y=2*(x/√2) - (x²)/2, which is √2 x - (x²)/2. Since x is a length (in units consistent with the square's side length), then √2 x is a length, but y is an area. Wait, hold on. This suggests something is wrong. Because √2 x is a length, and (x²)/2 is an area. Adding them would be mixing units. Therefore, there must be a mistake.Wait, no. Wait, in our shoelace formula, we computed the area as 2t -t². But t is a coordinate in the unit square (from0 to1). Therefore, the area y is unitless (since the square has area1). But when we substitute t =x /√2, x is a length (AP). Wait, this is conflicting.Wait, actually, in the coordinate system where the square has side length1, the distance from A(0,0) to P(t,t) is t√2. So AP =x = t√2. Therefore, t =x /√2. But in the coordinate system, the area computed via shoelace formula is in the unit of the coordinate system. Since the entire square has area1, then the shoelace formula gives area in the same unit. Therefore, when we computed y=2t -t², this is already in the unit area (relative to the square). So substituting t =x /√2 into y=2t -t² gives y=2*(x /√2) - (x /√2)^2 = (2x)/√2 -x² /2=√2 x -x² /2. Therefore, in terms of the square's area (which is1), this expression gives the correct proportion. Since x is a length (ranging up to√2 /2 to keep E on CD), then y=√2 x -x² /2 is the area in terms of x.But let's check the units again. If the square has side length1, then area is in [length]^2. So y should have units [length]^2. But √2 x is [length], and x² /2 is [length]^2. Therefore, we are adding a [length] term to a [length]^2 term, which is not possible. Therefore, there must be a mistake.Wait, this inconsiderate. Wait, actually, in the shoelace formula, when we computed y=2t -t², t is a coordinate in the unit square (each side length1). Therefore, t is dimensionless. Therefore, 2t -t² is dimensionless, and since the entire square has area1, the shoelace formula actually computes the area as a fraction of the total square. Wait, no. Wait, no. If the coordinates are in a unit square, then the shoelace formula computes the actual area. For example, when t=0.5, shoelace gives area=2*0.5 -0.5²=1 -0.25=0.75, which is correct (3/4 of the square). Therefore, in that case, the formula y=2t -t² computes the actual area (since the square is unit area). But when we express t in terms of x, which is a length (AP), then we have to ensure that x is in the same unit. Since in the unit square, AP is a distance from A(0,0) to P(t,t), which is t√2. Therefore, x= t√2. So t= x /√2. Therefore, substituting into y=2t -t² gives y=2*(x /√2) - (x /√2)^2= (2x)/√2 -x² /2=√2 x -x² /2. However, this expression combines terms with x (a length) and x² (an area). But since the entire square has area1, x is a length where the maximum x is√2 (diagonal), but when restricted to E on CD, x is up to√2/2. But if we consider the square to have side length1, then x is in [0,√2/2], and the area y=√2 x -x² /2 is in the units of the square's area (which is1). For example, when x=√2/2, y=√2*(√2/2) - ( (√2/2)^2 )/2= (2/2) - ( (2/4))/2=1 - (0.5)/2=1 -0.25=0.75, which is correct (as previously calculated). So even though the expression combines terms with x and x², in the context where the square has area1, it's acceptable because x is expressed in the same unit as the square's side. Therefore, the answer y=√2 x - (1/2)x² is correct.Therefore, the relationship between y and x is y=√2 x - (1/2)x².Final AnswerThe relationship between ( y ) and ( x ) is (boxed{y = sqrt{2}x - dfrac{1}{2}x^2}).</think>Given a square (ABCD) with side length 1, point (P) is on the diagonal (AC) such that (AP = x), and point (E) is on (CD) such that (PE perp PB). We need to find the relationship between the area (y) of the pentagon (ABPED) and (x).1. Coordinate System Setup: - Assign coordinates: (A(0,0)), (B(1,0)), (C(1,1)), (D(0,1)). - Point (P) on (AC) has coordinates ((t, t)) where (t) is a parameter. The distance (AP = x) implies (t = frac{x}{sqrt{2}}).2. Finding Coordinates of (E): - Point (E) is on (CD), so its coordinates are ((s, 1)). - Vectors (PE) and (PB) must be perpendicular. Using the dot product condition: [ (s - t, 1 - t) cdot (1 - t, -t) = 0 ] - Solving the dot product equation, we find (s = 2t). Therefore, coordinates of (E) are ((2t, 1)).3. Area Calculation Using Shoelace Formula: - Coordinates of the pentagon (ABPED): (A(0,0)), (B(1,0)), (P(t, t)), (E(2t, 1)), (D(0,1)). - Applying the shoelace formula: [ text{Area} = frac{1}{2} left| 0 + t + (t - 2t^2) + 2t + 0 - (0 + 0 + 0.5t^2 + 0 + 0) right| ] Simplifying, we get: [ text{Area} = frac{1}{2} |4t - 2t^2| = 2t - t^2 ]4. Substituting (t = frac{x}{sqrt{2}}): - Converting (t) back to (x): [ y = 2 left( frac{x}{sqrt{2}} right) - left( frac{x}{sqrt{2}} right)^2 = sqrt{2}x - frac{1}{2}x^2 ]Thus, the relationship between (y) and (x) is (boxed{y = sqrt{2}x - dfrac{1}{2}x^2}).]

🔑:2. y=-frac{1}{2} x^{2}+sqrt{2} xleft(0<x<frac{sqrt{2}}{2}right).As shown in Figure 4, draw PG perp BC and PH perp CD, with G and H being the feet of the perpendiculars. Then quadrilateral PGCH is a rectangle.Since CP is the angle bisector of angle BCD, we have PG = PH.Therefore, quadrilateral PGCH is a square.Since angle BPE = angle GPH = 90^{circ}, we have angle BPG = angle EPH.Thus, triangle BPG cong triangle EPH.Hence, S_{triangle BPG} = S_{triangle EPH}.At this point, S_{text{quadrilateral } BPEC} = S_{text{square } PGCH}.In square ABCD, it is easy to see that AC = sqrt{2}, and PC = sqrt{2} - x.Therefore, S_{text{square } PGCH} = frac{1}{2}(sqrt{2} - x)^{2}.Thus, y = S_{text{square } ABCD} - S_{text{quadrilateral } BPEC} = 1 - frac{1}{2}(sqrt{2} - x)^{2} = -frac{1}{2} x^{2} + sqrt{2} xleft(0 < x < frac{sqrt{2}}{2}right).

🧠:[<think>Okay, let's see. I need to figure out Julio's age based on the information given. The problem says:1. The sum of Zipporah's age and Dina's age is 51.2. The sum of Julio's age and Dina's age is 54.3. Zipporah is 7 years old.And the question is: How old is Julio?Alright, let me break this down step by step. First, let's note down the given information in terms of equations. That usually helps me visualize the relationships between the ages.Let me assign variables to each person's age to make it easier. Let's say:- Let Z be Zipporah's age.- Let D be Dina's age.- Let J be Julio's age.According to the problem:1. Z + D = 512. J + D = 543. Z = 7So we need to find J. Since we have two equations involving D, and we know Z's age, maybe we can first find D's age using the first equation and then use that to find Julio's age with the second equation.Starting with the first equation: Z + D = 51. If Z is 7, then substituting that in:7 + D = 51To solve for D, subtract 7 from both sides:D = 51 - 7D = 44Okay, so Dina is 44 years old. Now, moving on to the second equation: J + D = 54. We already know D is 44, so substitute that in:J + 44 = 54To find J, subtract 44 from both sides:J = 54 - 44J = 10Therefore, Julio is 10 years old. Let me double-check to make sure I didn't make any mistakes.First, Zipporah is 7, Dina is 44. Their combined ages: 7 + 44 = 51. That matches the first statement. Then Julio is 10, Dina is 44. Their combined ages: 10 + 44 = 54. That matches the second statement. Both equations check out, so the answer should be correct.Wait a second, just to ensure there's no oversight. Let me go through the steps again.1. From the first equation: Z + D = 51. Z is given as 7. So 7 + D = 51. Subtract 7: D = 44. Correct.2. From the second equation: J + D = 54. Since D is 44, substitute: J + 44 = 54. Subtract 44: J = 10. That seems right.No, everything looks consistent. Both sums add up correctly. So I think that's the right answer. Julio is 10 years old.Final AnswerJulio is boxed{10} years old.</think>Given the problem, we need to determine Julio's age based on the following information:1. The sum of Zipporah's age and Dina's age is 51.2. The sum of Julio's age and Dina's age is 54.3. Zipporah is 7 years old.We start by assigning variables to each person's age:- Let ( Z ) be Zipporah's age.- Let ( D ) be Dina's age.- Let ( J ) be Julio's age.From the first equation, we have:[ Z + D = 51 ]Given ( Z = 7 ), we substitute this value into the equation:[ 7 + D = 51 ]Solving for ( D ):[ D = 51 - 7 = 44 ]Next, using the second equation:[ J + D = 54 ]We substitute ( D = 44 ) into the equation:[ J + 44 = 54 ]Solving for ( J ):[ J = 54 - 44 = 10 ]Thus, Julio is (boxed{10}) years old.,<think>Okay, let's see. The problem is about figuring out Julio's age based on the information given. Let me read it again to make sure I understand all the parts.So, the first sentence says, "The sum of Zipporah's age and Dina's age is 51." That means if I add Zipporah's age and Dina's age together, it equals 51. Let me note that down: Z + D = 51.Then the second part says, "The sum of Julio's age and Dina's age is 54." So, adding Julio's age and Dina's age gives 54. I'll write that as: J + D = 54.The third piece of information is that Zipporah is 7 years old. So, Z = 7. The question is asking for Julio's age, which is J.Alright, so we need to find J. Let's see. Since we know Z is 7, maybe we can find Dina's age first from the first equation, and then use that to find Julio's age from the second equation.Starting with the first equation: Z + D = 51. If Z is 7, then substituting that in gives 7 + D = 51. To find D, I can subtract 7 from both sides: D = 51 - 7. Let me do that subtraction. 51 minus 7 is 44. So Dina is 44 years old.Now that we know Dina's age, we can use the second equation: J + D = 54. Substituting D with 44, we get J + 44 = 54. To solve for J, subtract 44 from both sides. 54 minus 44 is 10. So Julio is 10 years old.Let me check if that makes sense. If Zipporah is 7 and Dina is 44, together they are 51, which matches the first statement. Then Julio is 10, so adding Dina's 44 gives 54, which matches the second statement. Everything seems to add up correctly. I don't see any mistakes in the calculations. So Julio must be 10 years old.Final AnswerJulio is boxed{10} years old.</think>Given that the sum of Zipporah's age and Dina's age is 51, we can write the equation:[ Z + D = 51 ]We are also told that the sum of Julio's age and Dina's age is 54:[ J + D = 54 ]Zipporah's age is given as 7 years old:[ Z = 7 ]Substituting ( Z = 7 ) into the first equation:[ 7 + D = 51 ]Solving for ( D ):[ D = 51 - 7 = 44 ]Now, substituting ( D = 44 ) into the second equation:[ J + 44 = 54 ]Solving for ( J ):[ J = 54 - 44 = 10 ]Thus, Julio is (boxed{10}) years old.]

🔑:Since Zipporah is 7 years old and the sum of Zipporah's age and Dina's age is 51, then Dina is 51-7=44 years old.Since Dina is 44 years old and the sum of Julio's age and Dina's age is 54, then Julio is 54-44=10 years old.ANSWER: 10

🧠:[<think>Okay, so I have this problem here where I need to find the real number m such that the vectors a and b are parallel. The vectors are given as a = (m, 1) and b = (2, m - 1). Hmm, let me think. First, I remember that two vectors are parallel if one is a scalar multiple of the other. That means there exists some scalar k where a = k * b. So, if I write that out component-wise, each component of vector a should be k times the corresponding component of vector b. Let me write that down.So, for the x-component: m = k * 2And for the y-component: 1 = k * (m - 1)Alright, now I have two equations here with two variables: m and k. Maybe I can solve for k in one equation and substitute into the other. Let me try that.From the first equation, m = 2k. So, k = m / 2. Then plug that into the second equation. 1 = k * (m - 1) becomes 1 = (m / 2) * (m - 1). Hmm, so that gives me a quadratic equation. Let's expand that.Multiply both sides by 2 to eliminate the denominator: 2 * 1 = m * (m - 1)So, 2 = m^2 - mThen, rearranging terms: m^2 - m - 2 = 0Now, solving this quadratic equation. Let me see if it factors. Looking for two numbers that multiply to -2 and add to -1. Hmm, 1 and -2? Yes, 1 * (-2) = -2 and 1 + (-2) = -1. So, factors would be (m - 2)(m + 1) = 0.Wait, let me check that. (m - 2)(m + 1) = m^2 + m - 2m - 2 = m^2 - m - 2. Yes, that's right. So the solutions are m = 2 and m = -1.But wait, I need to make sure these solutions work in the original equations. Let me check both values.First, m = 2. Then vector a is (2, 1) and vector b is (2, 2 - 1) = (2, 1). So both vectors are (2, 1). Obviously, they are parallel because they are the same vector. So that works.Next, m = -1. Then vector a is (-1, 1) and vector b is (2, -1 - 1) = (2, -2). So vector b is (2, -2). Let's see if a is a scalar multiple of b. Let's check if there's a scalar k such that (-1, 1) = k*(2, -2). Looking at the x-component: -1 = 2k => k = -1/2. Then check the y-component: 1 = -2k. If k = -1/2, then -2k = -2*(-1/2) = 1. So yes, 1 = 1. So both components work. So m = -1 is also a valid solution.Therefore, both m = 2 and m = -1 are solutions. Wait, but the problem says "the real number m". Hmm, does that mean there are two solutions? Let me check the problem again."Given vectors a = (m,1) and b = (2, m−1), if a ∥ b, then the real number m is ______."It says "the real number m", which might suggest a single answer, but maybe both 2 and -1 are correct. Let me verify once more.Alternatively, maybe there's another way to check if vectors are parallel, by using the cross product. In two dimensions, the cross product (which gives the area of the parallelogram formed by the vectors) should be zero if they are parallel. The cross product for 2D vectors (x1, y1) and (x2, y2) is x1y2 - x2y1. So if that's zero, they are parallel.Let's compute that for vectors a and b. a = (m, 1), b = (2, m - 1). So cross product is m*(m - 1) - 2*1 = m(m - 1) - 2.Set that equal to zero: m(m - 1) - 2 = 0. Which is the same quadratic equation as before: m^2 - m - 2 = 0, leading to m = 2 and m = -1. So yes, same result.Therefore, the real numbers m are 2 and -1. But the problem says "the real number m". Maybe in Chinese or the original problem's phrasing, maybe they expect multiple answers? But the way it's written here in English, "the real number m is ______", perhaps expecting multiple answers. In Chinese problems, sometimes they use underscores for multiple answers. Let me check the original problem again."Given vectors a→=(m,1) and b→=(2,m−1), if a→ parallel to b→, then the real number m is ______."So the blank is plural, maybe? If it's expecting multiple answers, then both 2 and -1. But if the answer is a single number, maybe I made a mistake. Wait, let me check once again.Wait, cross product approach gives m^2 - m - 2 = 0, which factors to (m - 2)(m + 1) = 0, so m = 2 or m = -1. Both satisfy the condition. Therefore, the answer is m = 2 and m = -1. Since the problem didn't specify the number of answers, but in many cases, maybe it's two answers. But how to present them? In Chinese math problems, if there are multiple answers, sometimes they are separated by commas. Or maybe the problem is designed to have two answers. Let me check again.Original problem: "the real number m is ______". The blank may be intended for multiple answers. So maybe answer is 2 or -1, but in the format, we need to present both? Or perhaps there's an error here. Wait, let's check the cross product calculation again.Cross product is m*(m - 1) - 2*1 = m^2 - m - 2. Set to zero: m^2 - m - 2 = 0. Solutions m = [1 ± sqrt(1 + 8)] / 2 = [1 ± 3]/2. So 4/2 = 2 or -2/2 = -1. Correct. So both solutions are valid.Therefore, the answer should be m = 2 or m = -1. Since the problem says "the real number m", maybe they expect both. Therefore, the answer is 2 and -1. But how to present this in boxed notation? If two answers, maybe boxed{2} and boxed{-1}, but perhaps as a list. But the original problem didn't specify. Maybe the problem is from a source where multiple answers are expected. However, sometimes, when you think you have two answers, but one might be extraneous. Let me verify both solutions again.First, m = 2: a = (2,1), b = (2,1). So yes, parallel.Second, m = -1: a = (-1,1), b = (2, -2). So vector a is (-1,1), vector b is (2,-2). Then, check if one is scalar multiple. Let's see, if we multiply b by -0.5: 2*(-0.5) = -1, -2*(-0.5)=1. So yes, a = -0.5*b. So they are parallel. So both correct.Therefore, the answer should be both 2 and -1. However, if the problem is expecting multiple answers, perhaps they should be presented as two separate boxed numbers. But in Chinese math problems, sometimes they write multiple answers separated by commas inside one box. Maybe like boxed{2} and boxed{-1}, but I'm not sure. Alternatively, the problem might have a typo or expects only one answer. Wait, but both solutions are correct. Let me check once more.Alternatively, is there a case where m - 1 could lead to division by zero or something? Let's see. If m = 2, then vector b is (2,1). If m = -1, vector b is (2, -2). No division involved here. So both solutions are valid. Therefore, the answer is m = 2 and m = -1. But the problem says "the real number m is _____". Maybe it's expecting a set, like boxed{2} text{ and } boxed{-1}. But in the format requested, the user says "put your final answer within boxed{}". So likely, if there are two answers, they might need to be both in the box, perhaps separated by commas. Let me check the user's instruction: "put your final answer within boxed{}". So the final answer should be within a single box. But how to represent two numbers in one box? Maybe as a list: boxed{2} and boxed{-1}. But not sure. Wait, maybe the original problem in Chinese has only one answer. Let me check the equations again. Wait, maybe I made a mistake in the cross product method.Wait, cross product for vectors (x1,y1) and (x2,y2) is x1y2 - x2y1. So for a = (m,1) and b = (2, m-1):x1 = m, y1 = 1x2 = 2, y2 = m -1So cross product is m*(m -1) - 2*1 = m^2 - m -2. Set equal to zero: m^2 - m -2 =0. Solutions m=(1±√(1 + 8))/2=(1±3)/2=2 or -1. So correct.Therefore, two solutions. If the problem expects multiple answers, they should both be there. But given the translation, maybe in the original problem, the answer is two numbers. So I will present both in the box separated by commas. So boxed{2} and boxed{-1} but in one box. But how? Maybe boxed{2} boxed{-1}? But that's two boxes. Wait, in Chinese exams, sometimes they have multiple blanks, but in this case, the user instruction says "put your final answer within boxed{}". So perhaps the problem expects two answers, but the user wants both in one box. Maybe inside the box: boxed{2 text{ and } -1} or boxed{2, -1}. Since mathematical answers often use commas to separate multiple solutions. So probably boxed{2} and boxed{-1} if allowed multiple boxes, but user says "within boxed{}", so single box. So I'll go with boxed{2} and boxed{-1} but since the instructions specify a single box, perhaps boxed{2, -1}. Alternatively, check if the problem is from a source where only one answer is expected, but according to the math, both are correct. Therefore, to comply with the user's instruction, even though there are two answers, they need to be in one box. So I'll write boxed{2} and boxed{-1} but since the user says "within boxed{}", maybe separate them by commas inside a single box. I think that's acceptable. Therefore, the final answer is boxed{2} and boxed{-1} but given the instruction, perhaps boxed{2, -1}. Yes, that's likely the correct format here.Final AnswerThe real number ( m ) is boxed{2} and boxed{-1}.Wait, but maybe in the problem's context, they expect two answers, so as to put both in separate boxes. But the user specifies "put your final answer within boxed{}". Hmm. Looking back at the original problem statement: "the real number ( m ) is ______". The blank might be intended for two answers. In Chinese textbooks, sometimes they have multiple blanks, but here it's a single blank. However, according to the instructions, the assistant is supposed to put the final answer within boxed{}, which is a single box. So maybe the answer is two numbers in one box. For example, in some contexts, multiple answers are written inside the same box separated by commas. So perhaps boxed{2, -1}. Yes, that seems right. So to comply with the user's instruction of putting the final answer within a single box, I should present both solutions separated by a comma inside the box. Therefore, the final answer is boxed{2, -1}.Final AnswerThe real number ( m ) is boxed{2} and boxed{-1}.Wait, now I'm confused because different platforms have different conventions. In many cases, if there are multiple solutions, they are written in the same box separated by commas. Given the user's instruction says "put your final answer within boxed{}", which implies a single box. Therefore, I should combine both answers into one box. Hence, boxed{2, -1}. But in the thought process above, I initially concluded with two separate boxes. However, strictly following the user's instruction, since they mentioned "within boxed{}" (singular), it's safer to put both answers in one box. Let me check examples in the problem statement. The original problem says "the real number m is ______". If it's expecting two numbers, the blank is plural. But in Chinese, sometimes the way the question is phrased might not explicitly indicate the number of answers. So, given that, and to follow the user's instruction accurately, even though there are two answers, they should be in a single box. Therefore, the correct formatting is boxed{2, -1}.Final Answerboxed{2, -1}But actually, in standard practice for multiple answers in a single box, commas are used. So I'll go with that. Hence, the final answer is boxed{2, -1}.Final Answerboxed{2, -1}</think>Given vectors (overrightarrow{a} = (m, 1)) and (overrightarrow{b} = (2, m - 1)), we need to find the real number (m) such that (overrightarrow{a} parallel overrightarrow{b}).Two vectors are parallel if one is a scalar multiple of the other. This means there exists a scalar (k) such that (overrightarrow{a} = k overrightarrow{b}). Writing this component-wise, we have:[m = 2k quad text{and} quad 1 = k(m - 1)]Solving the first equation for (k), we get (k = frac{m}{2}). Substituting this into the second equation:[1 = left(frac{m}{2}right)(m - 1)]Multiplying both sides by 2:[2 = m(m - 1)]Rearranging gives the quadratic equation:[m^2 - m - 2 = 0]Factoring the quadratic equation:[(m - 2)(m + 1) = 0]Thus, the solutions are:[m = 2 quad text{or} quad m = -1]Verifying both solutions:- For (m = 2), (overrightarrow{a} = (2, 1)) and (overrightarrow{b} = (2, 1)), which are parallel.- For (m = -1), (overrightarrow{a} = (-1, 1)) and (overrightarrow{b} = (2, -2)), which are parallel since (overrightarrow{a} = -frac{1}{2} overrightarrow{b}).Both solutions are valid. Therefore, the real numbers (m) are (boxed{2}) and (boxed{-1}).However, considering the problem's requirement to present the final answer within a single box, the solutions are combined as:[boxed{2, -1}],<think>Okay, so I have this problem where there are two vectors, vector a is (m, 1) and vector b is (2, m-1). The question says that these vectors are parallel, and I need to find the real number m. Hmm, let me remember how vectors being parallel works. If two vectors are parallel, one must be a scalar multiple of the other. That means there exists some scalar k such that vector a = k times vector b, or vice versa. So, for each component of vector a, it should be equal to k times the corresponding component of vector b. Let me write that down.Given vector a = (m, 1) and vector b = (2, m - 1), if they are parallel, then there exists a scalar k where:m = k * 2 (from the x-components)1 = k * (m - 1) (from the y-components)So now I have two equations with two variables: m and k. I need to solve for m. Let me see how to do this. Maybe I can express k from the first equation and substitute it into the second equation.From the first equation: k = m / 2.Then substitute k into the second equation:1 = (m / 2) * (m - 1)So, 1 = (m(m - 1)) / 2. Let me multiply both sides by 2 to eliminate the denominator:2 = m(m - 1)Expanding the right side: 2 = m² - mBring all terms to one side to form a quadratic equation:m² - m - 2 = 0Now, I need to solve this quadratic equation. Let's see if it factors. The quadratic is m² - m - 2. Let me look for two numbers that multiply to -2 and add to -1. Hmm, factors of -2 could be 1 and -2. Let's check: 1 * (-2) = -2, and 1 + (-2) = -1. Yes, that works!So, factoring the quadratic: (m + 1)(m - 2) = 0Setting each factor equal to zero gives:m + 1 = 0 => m = -1orm - 2 = 0 => m = 2So, the solutions are m = -1 and m = 2. Wait, but I should check if both of these solutions work in the original equations. Sometimes when we solve equations, especially with substitutions, we might get extraneous solutions.First, let's check m = -1.If m = -1, then vector a = (-1, 1) and vector b = (2, -1 -1) = (2, -2). Let's see if vector a is a scalar multiple of vector b.Is there a k such that (-1, 1) = k*(2, -2)?For the x-component: -1 = 2k => k = -1/2Check the y-component: 1 = -2k. Plugging k = -1/2 gives 1 = -2*(-1/2) = 1. That works. So yes, m = -1 is valid.Now check m = 2.If m = 2, then vector a = (2, 1) and vector b = (2, 2 -1) = (2, 1). So vector a is (2,1) and vector b is (2,1). They are the same vector, so of course they are parallel (k = 1). So that's valid too.Wait, so both solutions are valid. So m can be either -1 or 2. But the problem says "the real number m is ______". Hmm, maybe it's expecting both answers? Let me check the problem again.The problem says: "if a parallel to b, then the real number m is ______." It doesn't specify if there are multiple answers. But maybe in the original problem, maybe it's a fill-in-the-blank with multiple answers. However, the way it's phrased here is "the real number m is...", which might suggest a single answer, but in reality, there are two possible answers.Wait, maybe I made a mistake in my thinking. Let me verify once again.The condition for two vectors being parallel is that one is a scalar multiple of the other. So, for vectors (m,1) and (2, m-1), we set up the ratios: m/2 = 1/(m - 1). That's another way to think about it: the components must be proportional.So, cross-multiplying gives m*(m - 1) = 2*1, which is m² - m = 2, so m² - m - 2 = 0, same as before. Factored as (m - 2)(m + 1) = 0, so m = 2 or m = -1. So yes, both are solutions. So the answer should be m = 2 or m = -1.But the problem is presented as "the real number m is _______", which in Chinese tests often allows multiple answers. But maybe in the original problem's context, the answer is two numbers. But since the user instruction here is to put the final answer within a box, perhaps they expect both values? How to format that?In Chinese math problems, sometimes they use commas or list them. Maybe boxed both answers. Let me check the initial problem statement again.Original problem: "Given vectors a = (m,1) and b = (2, m−1), if a is parallel to b, then the real number m is ______."Wait, in Chinese math problems, even if there are two answers, you have to write both, separated by commas. So, perhaps the answer is boxed{-1} and boxed{2}. But maybe in the system they expect both inside one box? I need to check the user's instruction. The user says: "put your final answer within boxed{}". So, probably need to put both answers in the box. But how?In some contexts, multiple answers are written as a list inside the box. For example, boxed{-1} and boxed{2}. But maybe they want both answers in one box. Let me see. Alternatively, maybe the problem is designed to have only one solution? Wait, let me check my calculations again.Wait, if m = 2, then vector a is (2,1) and vector b is (2, 1), which are the same vectors, so they are parallel. If m = -1, vector a is (-1,1), vector b is (2, -2), which are scalar multiples (k = -0.5). So both are correct. Therefore, there are two real numbers m. But the problem says "the real number m is...". Maybe it's a translation issue, perhaps in Chinese it's okay to have multiple answers. So in the blank, we write both -1 and 2.But since the user is asking for the answer in boxed notation, which is typically one box per answer. However, in some cases, multiple answers can be boxed together. For example, boxed{-1} and boxed{2}. Alternatively, if it's written as a set, boxed{-1, 2} or boxed{{-1, 2}}.But looking at the problem statement, since it's a fill-in-the-blank question expecting a real number, but there are two real numbers, so perhaps both must be given. Therefore, the correct answer is m = -1 and m = 2. So in boxed notation, perhaps separate boxes.But maybe the original answer is presented as two numbers, so I should check standard practices. In Chinese textbooks, when there are multiple answers, they are often separated by commas. For example, 2, -1. So, in boxed notation, maybe boxed{-1} and boxed{2} on separate lines, but the user's instruction says "put your final answer within boxed{}". So, maybe they want both inside a single box. Let me check online examples.Wait, in the problem statement, the user wrote: "then the real number m is ______". The Chinese original might have a blank expecting multiple answers. However, in Chinese math problems, if multiple answers, they are often written in the blank separated by commas. So, maybe in the box here, we can write both numbers separated by a comma inside a single box. For example, boxed{-1, 2} or boxed{2, -1}. Alternatively, sometimes they use curly braces to denote the set of solutions, like boxed{{-1, 2}}. But I need to check standard answer formatting. In many cases, for multiple solutions, they are presented as a list in the box. For example, on Zhihu or other Chinese platforms, multiple answers are separated by commas inside the box.Since the user hasn't specified, but the initial problem says "the real number m", maybe they expect multiple answers. But in the problem statement, since the vector components are given in terms of m, and there are two solutions, the answer should include both. So, I think the correct way is to present both answers in a single box, separated by commas. Therefore, the final answer is boxed{-1}, boxed{2} or boxed{-1, 2}. Let me verify once more.Wait, cross product method: for two vectors to be parallel, their cross product should be zero. In two dimensions, the cross product is represented as the determinant:|i   j   k||m   1   0||2  m-1 0|The cross product's magnitude is (m*(m - 1) - 1*2) = m² - m - 2. Setting this equal to zero gives m² - m - 2 = 0, same equation as before. Solutions m = 2 and m = -1. So yes, two solutions. Therefore, the answer is both -1 and 2.Therefore, in boxed notation, I should present both answers. Since the user's instruction says "put your final answer within boxed{}", but here we have two answers. If the system allows multiple boxes, then boxed{-1} and boxed{2}. However, in Chinese practice, sometimes multiple answers are written in one box separated by commas. For example, in many Chinese textbooks and exams, when a question expects multiple answers, you write them separated by commas in the blank. Therefore, perhaps here, to follow that, write boxed{-1, 2}.Yes, I think that's the best way. So the final answer is boxed{-1, 2}.Final AnswerThe real number ( m ) is boxed{-1, 2}.</think>Given vectors (overrightarrow{a} = (m, 1)) and (overrightarrow{b} = (2, m-1)), if (overrightarrow{a} parallel overrightarrow{b}), then there exists a scalar (k) such that (overrightarrow{a} = k overrightarrow{b}). This gives us the following system of equations:[m = 2k quad text{(from the x-components)}][1 = k(m - 1) quad text{(from the y-components)}]Solving for (k) from the first equation, we get (k = frac{m}{2}). Substituting this into the second equation:[1 = frac{m}{2}(m - 1)]Multiplying both sides by 2:[2 = m(m - 1)]Rearranging into a quadratic equation:[m^2 - m - 2 = 0]Factoring the quadratic equation:[(m + 1)(m - 2) = 0]This gives the solutions (m = -1) and (m = 2). Verifying both solutions:- For (m = -1), (overrightarrow{a} = (-1, 1)) and (overrightarrow{b} = (2, -2)), which are scalar multiples with (k = -frac{1}{2}).- For (m = 2), (overrightarrow{a} = (2, 1)) and (overrightarrow{b} = (2, 1)), which are identical vectors.Thus, the real numbers (m) are (boxed{-1, 2}).]

🔑:Given vectors overrightarrow{a}=(m,1) and overrightarrow{b}=(2,m-1), and knowing that overrightarrow{a} parallel overrightarrow{b}, we can deduce the relationship between their components. For two vectors to be parallel, the ratio of their corresponding components must be equal. Thus, we have:[frac{m}{2} = frac{1}{m-1}]Cross multiplying to eliminate the fractions, we get:[m(m-1) = 2 times 1]Expanding and simplifying the left side of the equation:[m^2 - m = 2]Rearranging the equation to bring all terms to one side:[m^2 - m - 2 = 0]This is a quadratic equation in the form of ax^2 + bx + c = 0. Solving this quadratic equation, we look for values of m that satisfy the equation. The solutions to a quadratic equation can be found using the formula m = frac{-b pm sqrt{b^2 - 4ac}}{2a}, where a = 1, b = -1, and c = -2. Substituting these values into the formula gives:[m = frac{-(-1) pm sqrt{(-1)^2 - 4(1)(-2)}}{2(1)} = frac{1 pm sqrt{1 + 8}}{2} = frac{1 pm sqrt{9}}{2} = frac{1 pm 3}{2}]Therefore, we have two possible solutions for m:[m = frac{1 + 3}{2} = 2 quad text{or} quad m = frac{1 - 3}{2} = -1]Hence, the real number m can be either 2 or -1. Thus, the final answer is encapsulated as:[boxed{2 text{ or } -1}]

🧠:[<think>Okay, so I need to find the number of intersection points formed by connecting 5 distinct points on line l₁ with 10 distinct points on line l₂ using line segments. The lines l₁ and l₂ are parallel, and no three segments intersect at the same point. Hmm, let's think about this step by step.First, since l₁ and l₂ are parallel, any line segment connecting a point on l₁ to a point on l₂ won't intersect another segment unless the two segments form a sort of "X" shape. That is, if two segments cross each other, they must connect different pairs of points such that their connecting lines cross between the two parallel lines.So, the key idea here is that each intersection is formed by two segments that cross each other. To count the number of such intersections, I need to determine how many such pairs of segments exist.But how exactly do these pairs form? Let's consider two segments AB and CD, where A and C are on l₁, and B and D are on l₂. These two segments will intersect if and only if the points are arranged in the order A, C on l₁ and B, D on l₂, but the segments cross each other. Wait, but since l₁ and l₂ are parallel, the segments AB and CD will cross if and only if the points are interleaved. That is, if on l₁ we have A before C, and on l₂ we have D before B. So, the ordering of the points on the two lines matters.Alternatively, another way to think about this is that choosing four points (two from l₁ and two from l₂) will determine exactly one intersection point. Because if you pick two points on each line, say A and B on l₁, and C and D on l₂, then connecting A to D and B to C will result in an intersection. Conversely, any intersection must come from such a combination of four points.Therefore, the number of intersection points is equal to the number of ways to choose two points from l₁ and two points from l₂. Because each such combination of four points (two on each line) will produce exactly one intersection when connected appropriately.So, mathematically, the number of intersections should be equal to the combination of 5 points taken 2 at a time multiplied by the combination of 10 points taken 2 at a time. That is, C(5,2) * C(10,2). Let me check if that makes sense.C(5,2) is the number of ways to choose two points on l₁, which is 10. C(10,2) is the number of ways to choose two points on l₂, which is 45. Multiplying these together gives 10 * 45 = 450. So, 450 intersection points. But wait, the problem states that no three line segments intersect at the same point. That condition is important because if three or more segments intersected at the same point, that would reduce the total number of intersections. However, the problem says that no three segments intersect at the same point, so each intersection is uniquely determined by two segments, which in turn are determined by two points on each line. Therefore, each pair of segments (determined by two points on each line) intersect exactly once, and no overlaps. Therefore, the total number is indeed 450.But let me verify this with a smaller example to make sure. Suppose l₁ has 2 points, A and B, and l₂ has 2 points, C and D. Then, connecting A to C, A to D, B to C, and B to D. The segments AC and BD intersect, and the segments AD and BC intersect. Wait, but in this case, if the lines are parallel, how do the segments intersect? Wait, hold on. If l₁ and l₂ are parallel, then the segments connecting them are non-crossing? Wait, no. Wait, if l₁ is above l₂, and the points are arranged in order. Let me visualize.Suppose l₁ is horizontal at the top, with points A and B from left to right. l₂ is horizontal below, with points C and D from left to right. Then, connecting A to C, A to D, B to C, B to D. The segments AC and BD cross each other, and segments AD and BC cross each other. Wait, but in reality, if you have two parallel lines, connecting leftmost on top to rightmost below and vice versa, those segments should cross in the middle. Wait, but with two points on each line, the two segments AC and BD cross each other once, and AD and BC cross each other once? Wait, no. Wait, in reality, if you connect A to D and B to C, those segments will cross each other once. Similarly, connecting A to C and B to D, those segments will cross each other once. So actually, for two points on each line, you get one intersection. Wait, no. Wait, hold on. Let me actually draw it mentally.If l₁ has points A and B from left to right, and l₂ has points C and D from left to right. Then, the segments AC and BD: do they cross? If A is connected to C (leftmost on top to leftmost on bottom) and B is connected to D (rightmost on top to rightmost on bottom), then those segments are parallel? Wait, no. Wait, lines AC and BD. Wait, if the lines l₁ and l₂ are parallel, then the segments AC and BD would not be parallel. Wait, maybe my mental image is off.Wait, no. If l₁ and l₂ are two horizontal lines, then connecting A (left on top) to D (right on bottom) and B (right on top) to C (left on bottom) would create two crossing segments. But if you connect A to C and B to D, those segments would not cross; they would be like two non-crossing lines. Wait, no. Wait, actually, in that case, connecting A to C and B to D would result in two segments that are like //, but if the lines are horizontal, then those would be two non-crossing diagonals. Whereas connecting A to D and B to C would result in an X shape between the lines.Wait a second, maybe my initial assumption is wrong. Maybe only when the connections are "crossing" in terms of the order, they intersect between the lines. So, if you connect A to C and B to D, they don't cross, but if you connect A to D and B to C, they do cross. Therefore, for two points on each line, there is exactly one intersection point when the segments cross each other. Thus, the number of intersections would be C(2,2)*C(2,2) = 1*1=1. But in reality, when you have two points on each line, you can form two segments that cross each other once. Wait, hold on. Let's see:If you have two points on the top line and two on the bottom. Each pair of segments that cross each other is formed by choosing one pair of points (A,B) on top and (C,D) on bottom, then connecting A to D and B to C. So, for each combination of two points on top and two on bottom, you get exactly one intersection. Therefore, the number of intersections is C(5,2)*C(10,2). So in the case of two points on each line, that would be C(2,2)*C(2,2)=1*1=1 intersection, which matches. Because if you have two points on each line, there is exactly one way to connect them such that the segments cross. Therefore, the formula seems to hold.Therefore, returning to the original problem, with 5 points on l₁ and 10 on l₂, the number of intersection points is C(5,2)*C(10,2). Let me compute that:C(5,2) = 10C(10,2) = 4510 * 45 = 450Therefore, the answer should be 450.But just to be absolutely sure, let me think again. Each intersection is uniquely determined by two segments, which in turn are determined by two points on l₁ and two points on l₂. Since the problem states that no three segments intersect at the same point, each intersection corresponds to exactly one such pair of segments. Hence, the total number of intersections is indeed the product of combinations C(5,2) and C(10,2), which is 10*45=450.Yes, that makes sense. Therefore, the number of intersection points is 450.Final AnswerThe number of intersection points is boxed{450}.</think>To determine the number of intersection points formed by connecting 5 distinct points on line ( l_1 ) with 10 distinct points on line ( l_2 ) using line segments, we need to consider how these segments intersect. Each intersection is formed by two line segments that cross each other. For two segments to intersect, they must connect pairs of points such that the points on each line are interleaved. This means choosing 2 points from ( l_1 ) and 2 points from ( l_2 ), which will form exactly one intersection point. The number of ways to choose 2 points from ( l_1 ) is given by the combination ( binom{5}{2} ), and the number of ways to choose 2 points from ( l_2 ) is given by ( binom{10}{2} ). The total number of intersection points is the product of these two combinations:[binom{5}{2} times binom{10}{2}]Calculating these values:[binom{5}{2} = frac{5 times 4}{2 times 1} = 10][binom{10}{2} = frac{10 times 9}{2 times 1} = 45]Multiplying these results gives:[10 times 45 = 450]Thus, the number of intersection points between the line segments is (boxed{450}).,<think>Okay, let's try to solve this problem step by step. So, we have two parallel lines, l₁ and l₂. Line l₁ has 5 distinct points, and line l₂ has 10 distinct points. We need to connect the points on l₁ with those on l₂ using line segments, and assuming no three line segments intersect at the same point, find the number of intersection points between these line segments.Hmm, first, I remember that when you connect points on two parallel lines with line segments, the intersections occur between these segments. Since the lines are parallel, the segments can't intersect on the lines themselves; all intersections must be between segments in the space between the lines.But how do we calculate the number of intersections? Let me think. Maybe it's related to how the segments cross each other. If two segments cross, they must form a sort of "X" shape. So, each intersection is formed by two segments that cross each other. To have two segments cross each other, the endpoints must be arranged in a certain way. Specifically, if we have two points on l₁, say A and B, and two points on l₂, say C and D, then the segments AC and BD will intersect if the points are ordered A, B on l₁ and D, C on l₂. Wait, but since l₁ and l₂ are parallel, the order of the points on each line matters. Let me visualize this.Suppose l₁ is horizontal at the top, with points A1, A2, A3, A4, A5 from left to right. l₂ is horizontal at the bottom, with points B1, B2, ..., B10 from left to right. If we connect A1 to B3 and A2 to B2, these two segments will cross each other somewhere between the lines. Similarly, if we have other connections, their crossings depend on the ordering of the endpoints.So, maybe the key is that an intersection occurs whenever two line segments connect four distinct points in such a way that their connections cross each other. That is, for any four points: two on l₁ and two on l₂, the segments connecting them will intersect if the pairs are "crossed" in order.Therefore, the number of intersection points should be equal to the number of such crossed pairs. So, for each combination of two points on l₁ and two points on l₂, there's exactly one intersection between the two segments if the connections cross. Therefore, the total number of intersections would be the number of ways to choose two points from l₁ and two points from l₂, multiplied by the number of crossings per such combination. But wait, for each set of two points on l₁ and two on l₂, how many intersections do they contribute?Let me think. Suppose we pick two points on l₁: let's say A and B (from l₁), and two points on l₂: C and D. If we connect A to C and B to D, the segments are AC and BD. If we connect A to D and B to C, the segments are AD and BC. In the first case, if the points are ordered A, B on l₁ and C, D on l₂, then AC and BD are parallel? Wait, no, they are not parallel. Wait, but if the lines l₁ and l₂ are parallel, then all the segments are non-parallel? No, the segments themselves are not parallel because they connect different points. Wait, but actually, two segments connecting points on two parallel lines can still be non-parallel. For example, connecting A1 to B1 and A2 to B2 are parallel if the lines are horizontal, but connecting A1 to B2 and A2 to B1 would cross each other.Wait, actually, no. If l₁ and l₂ are parallel, then segments connecting points on them can only be parallel if their slopes are the same. But since the lines are parallel, the slope between any two points would depend on the horizontal and vertical distance between the lines. Wait, maybe this is getting too complicated. Let me step back.Alternatively, perhaps each pair of segments that cross corresponds to a unique quadrilateral formed by two points on l₁ and two points on l₂. In such a quadrilateral, the two diagonals intersect. So, if we have four points: two on the top line (l₁) and two on the bottom line (l₂), then the diagonals of the quadrilateral (connecting A1 to B2 and A2 to B1) will intersect. So, each such quadrilateral contributes exactly one intersection point. Hence, the number of intersection points is equal to the number of such quadrilaterals.Therefore, the number of intersection points would be the combination of choosing 2 points from l₁ and 2 points from l₂. Since each such choice of 2 points on each line defines a quadrilateral, whose diagonals intersect once. So, the formula would be C(5, 2) * C(10, 2), where C(n, k) is the combination function.Calculating that, C(5,2) is 10 and C(10,2) is 45. So, multiplying them gives 10 * 45 = 450. Therefore, is the answer 450?Wait, but I need to verify this. Let me think again. If I have two lines with points, connecting every pair, then the number of intersection points inside is the number of convex quadrilaterals you can form, right? Because each intersection is formed by two crossing diagonals of a quadrilateral.So, to form a quadrilateral, you need two points on each line. Then, the diagonals of this quadrilateral intersect at one point. Since the lines are parallel, the quadrilateral is a trapezoid, but the diagonals of a trapezoid intersect. However, in a trapezoid, the diagonals do intersect, but in the case of a convex quadrilateral, they always intersect once. So, each quadrilateral gives exactly one intersection point. Therefore, the number of intersection points is equal to the number of such quadrilaterals, which is C(5,2)*C(10,2). So, 10*45=450.But wait, let me make sure there are no overcounting or undercounting. Suppose three segments intersect at the same point. The problem states that no three line segments intersect at the same point, so each intersection is uniquely determined by exactly two segments. Therefore, each intersection corresponds to exactly one pair of segments, which in turn corresponds to exactly one quadrilateral (i.e., two points on each line). Therefore, the total number of intersections is indeed C(5,2)*C(10,2).So, 5 choose 2 is 10, and 10 choose 2 is 45. Multiplying gives 450. So, the answer should be 450.But let me think again. Suppose we have two lines with m and n points. Then, the number of intersection points of the connecting segments is C(m, 2)*C(n, 2). Is that a standard formula?Yes, actually, I recall that when you have two parallel lines with m and n points, the number of intersection points formed by connecting all the points is indeed C(m, 2)*C(n, 2). Each intersection comes from choosing two points on each line, forming a quadrilateral whose diagonals intersect once. Since no three segments intersect at the same point (as given in the problem), all intersections are unique. Therefore, the formula holds.Therefore, substituting m=5 and n=10, we get C(5,2)*C(10,2)=10*45=450. So, the answer is 450. That seems correct. But let me check with a smaller example to be sure.Suppose l₁ has 2 points and l₂ has 2 points. Then, connecting them, there is only one intersection point? Wait, no. If you have two points on each line, connecting them forms two segments. If you connect A1 to B1 and A2 to B2, those segments are parallel and don't intersect. If you connect A1 to B2 and A2 to B1, those segments cross each other. Wait, so depending on how you connect the points, the number of intersections can vary? Wait, but in the problem, we are connecting all the points. So, if l₁ has 2 points and l₂ has 2 points, then there are 2*2=4 segments? Wait, no. Each point on l₁ is connected to each point on l₂, so if l₁ has 2 and l₂ has 2, we have 2*2=4 segments. However, each segment connects one point on l₁ to one on l₂. So, the total number of segments is 2*2=4. But how many intersections do they have?Wait, each pair of segments that cross each other would correspond to an intersection. But in this case, how many intersections are there? Let's see. If we have two lines with two points each: l₁: A, B; l₂: C, D. Then, the segments are AC, AD, BC, BD. Now, AC and BD cross each other, and AD and BC cross each other. Wait, no. Let me draw it mentally. If A is connected to C and D, and B is connected to C and D. So, segments AC and BD: If A is left on l₁, B is right on l₁; C is left on l₂, D is right on l₂. Then, AC goes from top-left to bottom-left, BD goes from top-right to bottom-right. These two segments cross each other. Similarly, AD goes from top-left to bottom-right, and BC goes from top-right to bottom-left. These two also cross each other. So, in total, there are two intersection points. But according to the formula C(2,2)*C(2,2)=1*1=1. But that contradicts the actual number of intersections which is 2. Hmm, so something is wrong here.Wait, maybe my initial assumption is incorrect. Wait, but in the formula, C(2,2)*C(2,2)=1*1=1, but we have two intersections. So, that contradicts. Therefore, my formula is wrong. So, where is the mistake?Wait, hold on. If we have two lines with 2 points each, how many intersection points are formed when all possible segments are drawn. Let's think again. Each segment connects a point on l₁ to a point on l₂. So, there are 4 segments: AC, AD, BC, BD. Each pair of segments that are not sharing a common endpoint will intersect unless they are parallel. But in this case, which pairs intersect?Segments AC and BD: intersect once.Segments AD and BC: intersect once.So, two intersection points. But according to the formula C(2,2)*C(2,2)=1*1=1. So, discrepancy here. So, the formula is not correct.Wait, so why the discrepancy? Let me analyze.Each intersection is formed by two segments that connect four distinct points (two on each line). So, in the case of two points on each line, there are two ways to pair the points. The two different pairings cross each other. Wait, but in this case, the four segments are AC, AD, BC, BD.The intersections are between AC & BD and between AD & BC.Each of these intersections corresponds to choosing two segments that form a "cross". So, each intersection is determined by a pair of segments, each connecting a different pair of points.But how many such pairs of segments are there?The number of pairs of segments is C(4,2)=6. But not all pairs intersect. Some pairs share a common endpoint. So, how many intersecting pairs?In the case of two points on each line:Number of segment pairs: C(4,2)=6.Number of pairs sharing a common endpoint: Each segment has two endpoints, so for each endpoint, the number of pairs sharing that endpoint. For point A on l₁: connected to C and D. So, pairs (AC, AD). Similarly, point B: (BC, BD). Points C and D on l₂: (AC, BC), (AD, BD). So, total of 4 pairs that share an endpoint. Therefore, total number of intersecting pairs is 6 - 4 = 2. Each of these intersecting pairs intersects once. So, two intersection points. But according to the formula C(2,2)*C(2,2)=1, but here we have 2. So, the formula is wrong.Therefore, my initial reasoning was incorrect. Therefore, there's a mistake in the logic.Wait, so maybe the formula isn't simply C(m,2)*C(n,2). Then, what's the correct approach?Alternatively, maybe each intersection is determined by choosing two points on l₁ and two points on l₂, but the number of intersections per such quadruple is not 1. Wait, in the case of two points on each line, how many intersections do we get? As above, two intersections. But C(2,2)*C(2,2)=1, so perhaps for each quadruple (two points on each line), there are two intersections? But that can't be.Wait, no. If you have two points on each line, you have four points total. Connecting them in all possible ways gives two intersections. But each intersection corresponds to a different pairing. So, perhaps each quadruple can give rise to multiple intersections? Wait, but with four points, two on each line, you can only connect them in two different ways (forming two crossing segments). Wait, no. If you have two points on each line, there are two possible ways to connect them with two segments: either connect A1 to B1 and A2 to B2 (which are parallel and don't intersect) or connect A1 to B2 and A2 to B1 (which cross each other). But in reality, when you connect all possible points, you have all four segments: A1B1, A1B2, A2B1, A2B2. The intersections are between A1B2 and A2B1, and between A1B1 and A2B2? Wait, no. Wait, actually, in the case where you have two points on each line, if you draw all four segments, the segments A1B1 and A2B2 are parallel (if the lines are parallel), so they don't intersect. Similarly, A1B2 and A2B1 cross each other. So, in this case, there is only one intersection point. Wait, but in my previous thought, I thought there were two. Wait, maybe I made a mistake there.Wait, let me clarify. If l₁ has points A1 and A2 (left to right), and l₂ has points B1 and B2 (left to right). Then, the segments are:1. A1B1: connects top-left to bottom-left.2. A1B2: connects top-left to bottom-right.3. A2B1: connects top-right to bottom-left.4. A2B2: connects top-right to bottom-right.Now, segments A1B2 and A2B1 cross each other. Segments A1B1 and A2B2 are parallel (they both go from left to left and right to right), so they don't intersect. Segments A1B1 and A2B1 both start at B1, so they share an endpoint. Similarly, A1B1 and A1B2 share endpoint A1. So, the only pair of segments that cross are A1B2 and A2B1. So, only one intersection point. But earlier, I thought there were two. That was my mistake. So, actually, with two points on each line, the number of intersections is 1. Then, according to the formula C(2,2)*C(2,2)=1*1=1, which matches. Then, where did I get two intersections before?Wait, maybe I was confusing with another case. Let me confirm again. If you have two points on each line, connecting all possible segments, how many intersections are there? Only one, where A1B2 crosses A2B1. The other segments are either parallel or share an endpoint, so they don't intersect. Therefore, the formula C(2,2)*C(2,2)=1 is correct here.Therefore, going back to the original problem: if we have m points on l₁ and n points on l₂, then the number of intersection points is C(m,2)*C(n,2). Because each pair of points on l₁ and pair on l₂ defines a quadrilateral whose two crossing segments intersect once. And since the problem states that no three segments intersect at the same point, each intersection is counted exactly once. Therefore, for the original problem with m=5 and n=10, the number of intersections is C(5,2)*C(10,2)=10*45=450.But let me verify this with another example. Suppose l₁ has 3 points and l₂ has 3 points. Then, according to the formula, the number of intersections should be C(3,2)*C(3,2)=3*3=9. Let's check manually.Points on l₁: A1, A2, A3 (left to right). Points on l₂: B1, B2, B3 (left to right). All segments are A1B1, A1B2, A1B3, A2B1, A2B2, A2B3, A3B1, A3B2, A3B3. Now, how many intersections are there?Each intersection is between two segments that form a crossing. For example, A1B2 and A2B1 cross. Similarly, A1B3 and A2B2 cross, A1B3 and A3B1 cross, A2B3 and A3B2 cross, etc. Let's count them:1. A1B2 intersects A2B12. A1B3 intersects A2B23. A1B3 intersects A3B14. A2B3 intersects A3B25. A1B2 intersects A3B1Wait, is that right? Wait, actually, let me list all possible pairs of segments that cross.First, choose two points on l₁: A1 and A2. Then two points on l₂: B1 and B2. Segments A1B2 and A2B1 cross. That's 1.A1 and A2 with B1 and B3: segments A1B3 and A2B1 cross. 2.A1 and A2 with B2 and B3: segments A1B3 and A2B2 cross. 3.Similarly, A1 and A3 with B1 and B2: segments A1B2 and A3B1 cross. 4.A1 and A3 with B1 and B3: segments A1B3 and A3B1 cross. 5.A1 and A3 with B2 and B3: segments A1B3 and A3B2 cross. 6.A2 and A3 with B1 and B2: segments A2B2 and A3B1 cross. 7.A2 and A3 with B1 and B3: segments A2B3 and A3B1 cross. 8.A2 and A3 with B2 and B3: segments A2B3 and A3B2 cross. 9.So, total 9 intersections. Which matches the formula C(3,2)*C(3,2)=3*3=9. Therefore, the formula works here.Therefore, the formula seems correct. Therefore, returning to the original problem with 5 points on l₁ and 10 points on l₂, the number of intersections is C(5,2)*C(10,2)=10*45=450. So, the answer should be 450.But just to make sure, let me think of another perspective. Each intersection is uniquely determined by four points: two on each line. For each such quartet, there is exactly one intersection point. Since no three segments intersect at the same point (given in the problem), each intersection is counted once. Therefore, the number of intersection points is the same as the number of ways to choose two points from l₁ and two from l₂, which is indeed C(5,2)*C(10,2)=450. Therefore, this seems solid.Therefore, I think the answer is 450.Final Answerboxed{450}</think>To determine the number of intersection points between line segments connecting 5 distinct points on line ( l_1 ) and 10 distinct points on line ( l_2 ), we need to consider how intersections are formed. Each intersection occurs when two line segments cross each other, which happens when the segments are formed by connecting two points on ( l_1 ) with two points on ( l_2 ) in a crossed manner.For any two points on ( l_1 ) and any two points on ( l_2 ), the line segments connecting them will intersect exactly once if the points are arranged in a crossed order. This means that each combination of two points from ( l_1 ) and two points from ( l_2 ) will contribute exactly one intersection point.The number of ways to choose 2 points from 5 on ( l_1 ) is given by ( binom{5}{2} ), and the number of ways to choose 2 points from 10 on ( l_2 ) is given by ( binom{10}{2} ). Multiplying these values gives the total number of intersection points.Calculating these combinations:[binom{5}{2} = frac{5 times 4}{2 times 1} = 10][binom{10}{2} = frac{10 times 9}{2 times 1} = 45]Multiplying these results:[10 times 45 = 450]Thus, the number of intersection points between the line segments is (boxed{450}).]

🔑:(12) 450 Hint: Any intersection point of two line segments, with 4 endpoints, two on l_{1} and two on l_{2}, forms a quadrilateral with these endpoints, and this intersection point is the intersection of the diagonals of the quadrilateral. Therefore, any intersection point corresponds one-to-one with a quadrilateral that has two vertices on l_{1} and two vertices on l_{2}. Thus, the total number of intersection points is mathrm{C}_{5}^{2} mathrm{C}_{10}^{2}= 450.