我是一名四年级的小学生， 这题怎么做？---**Question Number:** 8 **Question Stem:** 如图, 把一个正方形分成面积相等的 4 个长方形, 其中周长最小的一个长方形的周长是2025, 则这个正方形的周长是____。 (As shown in the figure, a square is divided into 4 rectangles with equal areas. The perimeter of the smallest rectangle is 2025. What is the perimeter of this square?) **Diagram Description:** * **Type:** Geometric figure. * **Main Elements:** A large square divided into four smaller rectangles by lines. * **Layout:** * The square is divided by a horizontal line across the top half and a vertical line across the bottom half. * The top half is occupied by a single rectangle spanning the full width of the square. * The bottom half is divided into three rectangles side-by-side, spanning the full width of the square. The middle rectangle appears narrower than the two side rectangles, and the two side rectangles appear to have the same width. * There are 4 rectangles in total as described in the text. Based on the visual representation and the problem statement about equal areas, the arrangement is likely one large rectangle at the top covering half the area, and three rectangles at the bottom covering the other half of the area, each having an area equal to one-fourth of the total square area. The diagram shows the top rectangle covering the top half, and the bottom half being divided into three rectangles. This implies that the top rectangle has an area equal to half of the square's area, and the three bottom rectangles together have an area equal to the other half. Since all four rectangles have equal area, this contradicts the visual layout if strictly interpreted as areas. However, the text states "divided into 4 rectangles with equal areas". Looking closely at the lines, the top rectangle's height is likely half the square's side. The bottom half is then divided vertically. To get 4 equal areas, the top rectangle must be formed by dividing the square into 2 equal horizontal halves, and then one half is divided vertically into 3 parts. This doesn't result in 4 rectangles of equal area unless the top rectangle is further divided. Let's re-examine the diagram relative to the text. The text says "divided into 4 rectangles with equal areas". The diagram shows a square divided by a horizontal line and a vertical line below it. The horizontal line divides the square into a top rectangle and a bottom area. The bottom area is then divided by vertical lines into three rectangles. This results in a total of 1 (top) + 3 (bottom) = 4 rectangles. If these 4 rectangles have equal areas, and the total area is that of the square, then each rectangle has an area equal to 1/4 of the square's area. In the diagram, the top rectangle spans the full width and occupies the top part. The bottom part spans the full width and is divided into three vertical strips. This layout implies the top rectangle's height is equal to the combined heights of the three bottom rectangles if stacked vertically, or some relationship based on width if placed side-by-side as shown. Let's assume the diagram accurately represents the division into 4 equal-area rectangles. This means the top rectangle's height must be 1/2 of the square's side, and the bottom half is divided into 3 rectangles, meaning their combined area is 1/2 of the square's area. But they are side-by-side, meaning their total width is the side of the square, and their heights are the same (the remaining half of the square's side). For them to have equal area (1/4 of the square each), their widths must sum up to the square's side, and their heights must be 1/2 the square's side. The top rectangle has height 1/2 the square's side and width equal to the square's side, giving an area of (1/2 * side) * side = 1/2 * side^2. This is half the square's area, not 1/4. Let's reinterpret the diagram based on the text's statement of 4 equal area rectangles. The diagram must show a way to divide a square into 4 rectangles of equal area. The most common way to divide a square into 4 equal area rectangles is by dividing it into 2x2 grid, but the diagram doesn't show that. Another way is to divide it into 4 identical rectangles placed side by side (or stacked), but then it wouldn't be a square unless the individual rectangles are squares. Let's reconsider the provided diagram as the correct configuration that results in 4 equal area rectangles when starting from a square. The diagram shows a square divided into two horizontal parts, a top part and a bottom part. The top part is one rectangle. The bottom part is divided into three vertical strips (rectangles). This results in 4 rectangles in total. For them to have equal area, let the side of the square be S. The total area is S^2. Each rectangle has area S^2 / 4. Let the height of the top rectangle be h1 and its width be w1. w1 = S. So Area1 = h1 * S = S^2 / 4. This gives h1 = S/4. The bottom part has height h2 = S - h1 = S - S/4 = 3S/4. The bottom part is divided into three rectangles with widths w2, w3, w4 and height h2. So w2 + w3 + w4 = S and h2 = 3S/4. The areas are A2 = w2 * h2, A3 = w3 * h2, A4 = w4 * h2. For A2 = A3 = A4 = S^2/4, we need w2 * (3S/4) = S^2/4, w3 * (3S/4) = S^2/4, w4 * (3S/4) = S^2/4. This implies w2 = w3 = w4 = (S^2/4) / (3S/4) = S/3. So, according to the text's condition of equal areas and the diagram's structure, the square (side S) is divided into: - Top rectangle: height S/4, width S. Area = (S/4) * S = S^2/4. - Bottom three rectangles: each has height 3S/4 and width S/3. Area = (3S/4) * (S/3) = S^2/4. This configuration matches the text and the diagram, although the relative proportions in the diagram might not be perfectly accurate. In the diagram, the top rectangle looks taller than S/4, and the bottom rectangles look wider than S/3 relative to the remaining height. However, we must trust the condition of equal areas and the division method shown. Let's identify the dimensions of the 4 rectangles based on this analysis: - Rectangle 1 (Top): dimensions S and S/4. Perimeter = 2 * (S + S/4) = 2 * (5S/4) = 5S/2. - Rectangle 2 (Bottom Left): dimensions S/3 and 3S/4. Perimeter = 2 * (S/3 + 3S/4) = 2 * ((4S + 9S)/12) = 2 * (13S/12) = 13S/6. - Rectangle 3 (Bottom Middle): dimensions S/3 and 3S/4. Perimeter = 13S/6. - Rectangle 4 (Bottom Right): dimensions S/3 and 3S/4. Perimeter = 13S/6. Let's compare the perimeters: - Perimeter of top rectangle: 5S/2 = 2.5S - Perimeter of bottom rectangles: 13S/6 = 2.166...S The problem states that the perimeter of the *smallest* rectangle is 2025. Comparing 2.5S and 2.166...S, the smallest perimeter belongs to the three bottom rectangles. So, the perimeter of one of the bottom rectangles is 2025. 13S/6 = 2025 S = 2025 * 6 / 13 The question asks for the perimeter of the large square. Perimeter of the square = 4 * S. Perimeter of square = 4 * (2025 * 6 / 13) = (4 * 2025 * 6) / 13 = (24 * 2025) / 13. Let's check the calculation. 2025 * 24 = 2025 * (20 + 4) = 40500 + 8100 = 48600. Perimeter of square = 48600 / 13. 48600 / 13 = ? 48600 = 13 * 3000 + 9600 9600 = 13 * 700 + 500 500 = 13 * 30 + 110 110 = 13 * 8 + 6 So 48600 / 13 = 3000 + 700 + 30 + 8 + 6/13 = 3738 and 6/13. The result should likely be an integer or a simpler fraction. Let me re-examine the problem statement and diagram. Maybe the diagram represents a different division. What if the square is divided by a vertical line down the middle, and one half is divided horizontally into two rectangles, and the other half is divided horizontally into two rectangles? That would be 4 equal rectangles, each with dimensions S/2 by S/2, forming 4 smaller squares. The perimeter of each is 2 * (S/2 + S/2) = 2S. The perimeter of the large square is 4S. In this case, all 4 rectangles are squares and have the same perimeter. This doesn't fit the description of "the smallest rectangle". Let's look at the diagram again. It shows a horizontal line, dividing the square into two parts. The top part is one rectangle. The bottom part is divided by vertical lines into three rectangles. Let the side of the square be S. Let the height of the top rectangle be y1, and its width be x1. Since it spans the top, x1 = S. Let the height of the bottom section be y2, and it's divided into three rectangles with widths x2, x3, x4 and height y2. So x2 + x3 + x4 = S, and y1 + y2 = S. The areas are: A1 = x1 * y1 = S * y1 A2 = x2 * y2 A3 = x3 * y2 A4 = x4 * y2 We are given A1 = A2 = A3 = A4 = Area_square / 4 = S^2 / 4. From A1 = S * y1 = S^2 / 4, we get y1 = S/4. Then y2 = S - y1 = S - S/4 = 3S/4. So A2 = x2 * (3S/4) = S^2 / 4 => x2 = (S^2/4) / (3S/4) = S/3. A3 = x3 * (3S/4) = S^2 / 4 => x3 = S/3. A4 = x4 * (3S/4) = S^2 / 4 => x4 = S/3. Check: x2 + x3 + x4 = S/3 + S/3 + S/3 = 3S/3 = S. This is consistent. The dimensions of the four rectangles are: - Rectangle 1 (Top): width S, height S/4. Perimeter P1 = 2 * (S + S/4) = 2 * (5S/4) = 5S/2. - Rectangle 2 (Bottom Left): width S/3, height 3S/4. Perimeter P2 = 2 * (S/3 + 3S/4) = 2 * ((4S + 9S)/12) = 2 * (13S/12) = 13S/6. - Rectangle 3 (Bottom Middle): width S/3, height 3S/4. Perimeter P3 = 13S/6. - Rectangle 4 (Bottom Right): width S/3, height 3S/4. Perimeter P4 = 13S/6. Comparing the perimeters: P1 = 5S/2 = 2.5S P2 = P3 = P4 = 13S/6 ≈ 2.1667S The smallest perimeter is P2 = P3 = P4 = 13S/6. We are given that the perimeter of the smallest rectangle is 2025. So, 13S/6 = 2025. S = 2025 * 6 / 13. The perimeter of the large square is 4S. Perimeter_square = 4 * (2025 * 6 / 13) = (4 * 6 * 2025) / 13 = (24 * 2025) / 13. 2025 is divisible by 3 (sum of digits 9), by 5 (ends in 5), by 9 (sum of digits 9). 2025 = 25 * 81 = 5^2 * 9^2 = 5^2 * 3^4. 24 = 2^3 * 3. So, (24 * 2025) / 13 = (2^3 * 3 * 5^2 * 3^4) / 13 = (2^3 * 3^5 * 5^2) / 13. There seems to be no common factor with 13. Let's check the calculation of 2025 * 24. 2025 * 24 = 48600. 48600 / 13. Let's perform long division: 48600 / 13 48 / 13 = 3 remainder 9 (13 * 3 = 39) 96 / 13 = 7 remainder 5 (13 * 7 = 91) 50 / 13 = 3 remainder 11 (13 * 3 = 39) 110 / 13 = 8 remainder 6 (13 * 8 = 104) So, 48600 / 13 = 3738 with remainder 6. The result is 3738 and 6/13. Is it possible that the diagram represents the dimensions differently? Let's assume the square has side S. Let's assume the division is as shown. Let the height of the top rectangle be x, and its width be S. Its area is Sx. Let the height of the bottom part be S-x. The bottom part is divided into three rectangles. Let their widths be w1, w2, w3, and their height be S-x. Areas are w1(S-x), w2(S-x), w3(S-x). Total area = Sx + w1(S-x) + w2(S-x) + w3(S-x) = S^2. Also, w1 + w2 + w3 = S. All 4 areas are equal, so Sx = w1(S-x) = w2(S-x) = w3(S-x) = S^2/4. From Sx = S^2/4, we get x = S/4. This confirms the earlier calculation for the height of the top rectangle. Then the height of the bottom rectangles is S-x = S - S/4 = 3S/4. w1(3S/4) = S^2/4 => w1 = (S^2/4) / (3S/4) = S/3. Similarly, w2 = S/3 and w3 = S/3. This again leads to the same dimensions. Rectangle 1: S by S/4. Perimeter 2(S + S/4) = 5S/2. Rectangles 2, 3, 4: S/3 by 3S/4. Perimeter 2(S/3 + 3S/4) = 2(4S/12 + 9S/12) = 2(13S/12) = 13S/6. Smallest perimeter is 13S/6. 13S/6 = 2025. S = 2025 * 6 / 13. Perimeter of square = 4S = 4 * (2025 * 6 / 13) = 48600 / 13. Let me double check the problem wording or common variations of this problem. This is a known type of problem. Let's see if I misinterpreted "smallest rectangle". In the diagram, the rectangle with dimensions S/3 and 3S/4 looks elongated (height > width, since 3/4 > 1/3). The top rectangle with dimensions S and S/4 looks wide (width > height, S > S/4). Maybe the term "smallest" refers to area, but the problem explicitly says "面积相等的 4 个长方形" (4 rectangles with equal areas). So "smallest" must refer to perimeter among these equal-area rectangles. Let's rethink the possibility that the diagram is misleading and there's another way to divide a square into 4 equal area rectangles that results in a different set of perimeters. What if the square is divided into a central rectangle surrounded by L-shaped regions, which are then further subdivided? No, the diagram shows a simple rectilinear division. What if the side lengths are in some simple ratio? Let the side of the square be S. The area is S^2. Each rectangle has area S^2/4. Consider a rectangle with sides a and b. Area = ab = S^2/4. Perimeter = 2(a+b). For a fixed area, the perimeter is minimized when the rectangle is a square (a=b). In our case, ab = S^2/4. If a = b, then a^2 = S^2/4, so a = S/2. The rectangle is a square with side S/2. Perimeter = 2(S/2 + S/2) = 2S. If all four rectangles were S/2 by S/2 squares, the total area would be 4 * (S/2)^2 = 4 * S^2/4 = S^2, which fits. In this case, the original square is divided into a 2x2 grid of squares. However, this doesn't fit the diagram and all four rectangles would have the same perimeter, so there wouldn't be a "smallest" one by perimeter. Let's go back to the dimensions S by S/4 and S/3 by 3S/4. Rectangle 1: sides S and S/4. Ratio of sides = S / (S/4) = 4. Rectangles 2, 3, 4: sides S/3 and 3S/4. Ratio of sides = (3S/4) / (S/3) = (3/4) * 3 = 9/4 = 2.25. Or the reciprocal ratio is 4/9. For a fixed area (S^2/4), the perimeter 2(a+b) is minimized when a and b are as close as possible (ratio close to 1). Comparing the ratios 4 and 2.25, the ratio 2.25 is closer to 1 than 4. So, the rectangle with sides S/3 and 3S/4 should have a smaller perimeter than the rectangle with sides S and S/4, given they have the same area. This confirms our calculation that 13S/6 < 5S/2. 13/6 ≈ 2.167 5/2 = 2.5 So the dimensions S/3 and 3S/4 are indeed for the rectangle(s) with the smallest perimeter among the four. Let's re-read the question carefully. "其中周长最小的一个长方形的周长是2025". Yes, "among them, the perimeter of the smallest rectangle is 2025". This confirms that 13S/6 = 2025. Let's recheck the calculation of S and the perimeter of the square. 13S/6 = 2025 S = 2025 * 6 / 13 Perimeter of square = 4S = 4 * (2025 * 6 / 13) = 24 * 2025 / 13. Let's reconsider the possibility of a mistake in interpreting the diagram or the problem. Could there be a simpler solution? Perhaps the side lengths of the rectangles are related in a simple way that makes the arithmetic easier. Let the side of the square be 1 unit for simplicity in finding the ratios of the perimeters. Then S=1. Rectangle 1: sides 1 and 1/4. Area 1/4. Perimeter 2(1 + 1/4) = 2(5/4) = 5/2. Rectangles 2, 3, 4: sides 1/3 and 3/4. Area (1/3)*(3/4) = 1/4. Perimeter 2(1/3 + 3/4) = 2((4+9)/12) = 2(13/12) = 13/6. Perimeter ratios are 5/2 : 13/6 = 15/6 : 13/6 = 15 : 13. The smallest perimeter is 13/6 relative to S=1. If the side of the square is S, the perimeters are 5S/2 and 13S/6. Smallest perimeter is 13S/6. Given 13S/6 = 2025. Perimeter of square = 4S. We need to find 4S. From 13S/6 = 2025, we have S = 2025 * 6 / 13. 4S = 4 * (2025 * 6 / 13) = 24 * 2025 / 13. Let's think about the numbers. 2025. It's a round number. 2025 = 25 * 81. Could there be a unit conversion or a hidden relationship? Let's check if 2025 is divisible by 13. 2025 / 13: 20 / 13 = 1 rem 7 72 / 13 = 5 rem 7 (13 * 5 = 65) 75 / 13 = 5 rem 10 (13 * 5 = 65) No, 2025 is not divisible by 13. Is it possible that the problem is designed such that the result 48600/13 is the intended answer? It's possible, but usually competition math problems have integer or simple fractional answers. Let me re-examine the image, especially the numbers and characters. Everything seems clear. "周长最小的一个长方形的周长是2025". Let's check for common errors or alternative interpretations of this type of problem. Sometimes the diagram can be misleading. But assuming the diagram and the text are consistent, our derivation of the dimensions seems correct based on the "4 equal area rectangles" condition and the shown division pattern. Let's check the perimeters again. P_top = 2(S + S/4) = 5S/2 P_bottom = 2(S/3 + 3S/4) = 2(4S/12 + 9S/12) = 2(13S/12) = 13S/6 Comparing 5/2 and 13/6. 5/2 = 15/6. 15/6 > 13/6. So 13S/6 is indeed the smallest perimeter. Let's try to work backwards from a potential integer answer for the square's perimeter. If Perimeter_square = P, then S = P/4. The smallest perimeter is 13S/6 = 13(P/4)/6 = 13P/24. We are given 13P/24 = 2025. P = 2025 * 24 / 13. This is the same calculation. Let's check if 2025 * 24 is divisible by 13. We found 2025 is not divisible by 13, and 24 is not divisible by 13. Their product might be, but we already did the division and got a remainder. Could the problem be phrased in a way that implies a different division? "把一个正方形分成面积相等的 4 个长方形" (divide a square into 4 rectangles of equal area). The diagram then shows how it is divided. Let's re-examine the number 2025. It ends in 25, so it is divisible by 25. 2025 = 25 * 81. 24 = 2^3 * 3. 13 is a prime number. Calculation: 24 * 2025 = 48600. 48600 / 13. Is it possible there is a typo in the number 2025? Let's assume the problem and diagram are correct and my calculations are correct. Then the answer is 48600/13. However, this seems like an unlikely intended answer. Let's reconsider the diagram. Is there any other interpretation? Could the bottom part be divided differently? No, it is clearly divided into three vertical strips. Could the top part be different? No, it's a single rectangle spanning the width. Let's look for similar problems online. "square divided into 4 equal area rectangles". Many results show the 2x2 grid. Some show a division into 1x4 (four identical rectangles in a row or column), or other configurations. The configuration shown in the diagram is less standard for "equal area" division, unless the dimensions are precisely as calculated (S by S/4 and S/3 by 3S/4). Let's verify the perimeter formula and area formula. These are standard. Let's verify the comparison of perimeters for equal area rectangles. For area A, perimeter is 2(a+b) where ab=A. Perimeter is minimized when a=b=sqrt(A). It increases as the ratio a/b deviates from 1. For area S^2/4: Rectangle 1: sides S and S/4. Ratio S / (S/4) = 4. Rectangles 2, 3, 4: sides S/3 and 3S/4. Ratio (3S/4) / (S/3) = 9/4 = 2.25. Since 2.25 is closer to 1 than 4, the perimeter of the 3S/4 by S/3 rectangle is indeed smaller than the perimeter of the S by S/4 rectangle. So, the smallest perimeter is 13S/6. 13S/6 = 2025. S = 2025 * 6 / 13. Perimeter of square = 4S = 4 * 2025 * 6 / 13 = 24 * 2025 / 13 = 48600 / 13. Let's consider the possibility that 2025 is related to 13 in some way I missed, or maybe related to 24 or 6. 2025 = 25 * 81. 13 is prime. 6 = 2 * 3. 24 = 2^3 * 3. Is there a possibility that the smallest rectangle is oriented differently in the diagram? No, the diagram shows the orientation. Let's consider the possibility that the question meant "the perimeter of the rectangle with the smallest perimeter is 2025". This is what the sentence says. Could there be a simpler relationship between the perimeter of the smallest rectangle and the perimeter of the square? Perimeter_smallest = 13S/6 Perimeter_square = 4S Ratio = (13S/6) / (4S) = (13/6) / 4 = 13/24. So, Perimeter_smallest = (13/24) * Perimeter_square. Perimeter_square = Perimeter_smallest * (24/13). Perimeter_square = 2025 * (24/13). This calculation is exactly the same. Let's check if 2025 * 24 is a multiple of 13. 2025 = 2028 - 3. 2028 = 13 * 156. So 2025 = 13 * 156 - 3. 2025 * 24 = (13 * 156 - 3) * 24 = 13 * 156 * 24 - 3 * 24 = 13 * 156 * 24 - 72. 72 = 5 * 13 + 7. So 2025 * 24 = 13 * (156 * 24) - (5 * 13 + 7) = 13 * (156 * 24 - 5) - 7. This confirms that 2025 * 24 is not divisible by 13, and the remainder is related to -7 mod 13, which is 6 mod 13. Our previous calculation 48600 / 13 = 3738 remainder 6 is correct. Perhaps the problem expects the answer as a fraction, or maybe there is a typo in the number 2025. If, for example, the perimeter was 202.5, or 202.8, or something related to 13. If 13S/6 = 2028, then S = 2028 * 6 / 13 = (13 * 156) * 6 / 13 = 156 * 6 = 936. Then Perimeter_square = 4S = 4 * 936 = 3744. This would be a nice integer answer. 2028 is close to 2025. Maybe 2025 was intended to be 2028? Or maybe the original value was in a different unit, and 2025 is the result after some conversion? Unlikely in a math competition problem without mentioning units. Let's assume the number 2025 is correct as given. Then the calculated perimeter of the square is 48600/13. Since the request is to extract the content and output it, I should output the question as it is, including the blank for the answer. The calculation steps are internal thinking process to understand the problem and determine the value that goes into the blank. The question asks for "这个正方形的周长是____". This is a fill-in-the-blank question. Based on the analysis, the value should be 48600/13. However, since it's a blank and might expect a simplified form or an integer, I should just extract the question text completely and accurately. Let's check if I missed any part of the image. No, just the question number, the question text, the blank space, and the diagram. There are no options provided. So, I will extract the question number and the question text as provided. I will also describe the diagram. Regarding the extraction content requirements: 1. Textual Information: * Question Stem: Yes, extracted. * Options: Not present. * Other Relevant Text: Question number 8, blank space, degree symbol (which is not a degree symbol, but the end of a sentence punctuation mark '。'). * Mathematical Formulas/Chemical Equations: Implicit in the problem description (area, perimeter formulas). Not explicitly written out as formulas in the text part. * Table Content: Not present. 2. Chart/Diagram Description: Yes, describe the geometric figure and its division. Let me structure the output. Question 8. Question Stem: 如图, 把一个正方形分成面积相等的 4 个长方形, 其中周长最小的一个长方形的周长是2025, 则这个正方形的周长是____。 (As shown in the figure, a square is divided into 4 rectangles with equal areas. The perimeter of the smallest rectangle is 2025. What is the perimeter of this square?) Diagram Description: The diagram shows a square. The square is divided by a horizontal line into a top rectangle and a bottom region. The bottom region is divided by two vertical lines into three rectangles. In total, there are four rectangles within the square. Based on the problem statement, these four rectangles have equal areas. I will output the Chinese text of the question stem, followed by an English translation for clarity in this thought process, but the final output should be based on the original language requirement which is plain text extraction. The requirement says "completely and accurately extract all content related to the question(s) from the image". It doesn't explicitly require translation, but asks for "structured plain text format". I will provide the original text. Final check on extraction requirements: "completely and accurately extract all content". "structured plain text format". I will output the question number, the question text (including the blank space and the punctuation), and the diagram description. Let's refine the diagram description to be purely descriptive of the visual elements and their relative positions/structure, without interpreting the implied equal areas unless explicitly stated in the diagram itself (which it is not). The text states the equal areas. So the diagram description should focus on the visual layout. Diagram Description: A square shape is shown. Inside the square, there is a horizontal line in the upper part, creating a rectangle above the line and a larger area below the line. The area below the horizontal line is further divided by two vertical lines into three smaller vertical rectangles arranged side-by-side. This results in a total of four rectangles within the original square: one large rectangle at the top and three smaller rectangles in the bottom row. This description accurately reflects the visual content of the diagram. The problem text provides the constraint about equal areas and the perimeter of the smallest rectangle. Let's consider the blank space. It's part of the content to be extracted. I will represent it as ____. The punctuation mark at the end of the sentence '。' should also be included. The question number '8.' is also part of the content. So, the final output should include: "8. " The Chinese sentence. The blank space "____". The punctuation mark "。". And the description of the diagram. The description of the diagram can be placed after the question stem. Let me double check the instructions. "Please completely and accurately extract all content related to the question(s) from the image I send, and output it in a structured plain text format." "For different sections such as the question stem, options, chart description, etc., please use clear paragraphs or markers for readability." Okay, I will use markers for different sections.