AMC 8 2012 Problem 25
AMC8 2012 年第 25 题
四边形与多边形★★★★☆
A square with area 4 is inscribed in a square with area 5, with one vertex of the smaller square on each side of the larger square. A vertex of the smaller square divides a side of the larger square into two segments, one of length a , and the other of length b . What is the value of ab ?
一个面积为 4 的正方形内接在一个面积为 5 的正方形内，小正方形的每个顶点分别落在大正方形的每条边上。小正方形的某个顶点将大正方形的一条边分成了 2 条线段，长度分别是 a 和 b，那么 ab 的值是多少？
- A.
- B.
- C.
- D.
- E.
正确答案 · Correct Answer
C
解析 · Solution
解法1
常规解法 · 四个直角三角形
大方与小方之间是 4 个全等直角三角形，总面积 5-4=1，每个 。每个三角形两直角边为 、，面积 ，故 。
巧解 · 和平方
大方边长为 a+b，；小方边长为 ，。两式相减
故 。
Solution 1
Elementary · four right triangles
The region between the squares is four congruent right triangles with total area 5-4=1, so each has area . With legs and , , hence .
Slick · square of the sum
The larger square has side a+b with , and the smaller has side with . Subtracting,
so .
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