AMC 8 2006 Problem 5
AMC8 2006 年第 5 题
四边形与多边形★★☆☆☆
Points A, B, C and D are midpoints of the sides of the larger square. If the larger square has area 60, what is the area of the smaller square?
点 A，B，C 和 D 是一个大正方形的各边的中点。若大正方形的面积为 60，那么小正方形的面积是多少？
- A.15
- B.20
- C.24
- D.30
- E.40
正确答案 · Correct Answer
D
解析 · Solution
Solution 1
Drawing segments and , the number of triangles outside square is the same as the number of triangles inside the square. Thus areas must be equal so the area of is half the area of the larger square which is .
If the side length of the larger square is , the side length of the smaller square is . Therefore the area of the smaller square is , half of the larger square's area, . Thus, the area of the smaller square in the picture is .
The smaller square forms four congruent isosceles right triangles whose legs measure half of the larger square's side length. Since the larger square has an area of 60, its sides measure . Therefore, the legs of the right triangles measure . Now we can use the Pythagorean Theorem to find the length of the hypotenuses, or the smaller square's sides. We have Squaring this, the area of the smaller square is 30.
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