AMC 8 2019 Problem 2
AMC8 2019 年第 2 题
四边形与多边形★★☆☆☆
Three identical rectangles are put together to form rectangle ABCD , as shown in the figure below. Given that the length of the shorter side of each of the rectangles is 5 feet, what is the area in square feet of rectangle ABCD ?
如图，把三个全等矩形拼在一起形成矩形 ABCD，已知每个小的矩形的短边为 5 英尺，求矩形 ABCD 的面积是多少平方英尺？
- A.45
- B.75
- C.100
- D.125
- E.150
正确答案 · Correct Answer
E
解析 · Solution
Solution 1
We can see that there are 2 rectangles lying on top of the other and that is the same as the length of one rectangle. Now we know that the shorter side is 5, so the bigger side is 10, if we do . Now we get the sides of the big rectangle being 15 and 10, so the area is 150.
Using the diagram we find that the larger side of the small rectangle is 2 times the length of the smaller side. Therefore, the longer side is . So the area of the identical rectangles is . We have 3 identical rectangles that form the large rectangle. Therefore the area of the large rectangle is . ~
We see that if the short sides are 5, the long side has to be because the long side is equal to the 2 short sides and because the rectangles are congruent. If that is to be, then the long side of the BIG rectangle(rectangle ) is 10+5=15 because long side + short side of the small rectangle is 15. The short side of rectangle is 10 because it is the long side of the short rectangle. Multiplying 15 and 10 together gets us which is 150. ~
We see that the 2 rectangles lying on top of each other give us the height of the rectangle. Using what we know, we can find out that the 2 rectangles put together is a square. So, we can infer that the length of the rectangles is 10. Adding that to the width of the third rectangle which is 5, we get that the length of the rectangle is 15. Multiplying 10 and 15 gives us which is 150. ~
There are two rectangles lying on the side and one standing up. Given that one small side is 5, we can determine that two of the small sides make up a big side which means that the long side is equal to 10. The top side of the rectangle is made up of one small side and one long side, therefore the dimensions for the rectangle is 10x15. 10 multiplied by 15 is 150, hence the answer 150. ALL THESE SOLUTIONS ARE VERY, VERY SIMILAR
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