AMC8 2012

AMC8 2012 · Q25

AMC8 2012 · Q25. It mainly tests Area & perimeter, Coordinate geometry.

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的正方形内接于一个面积为5的正方形，小正方形的每个顶点位于大正方形的一条边上。小正方形的一个顶点将大正方形的一条边分成两段，长分别为$a$和$b$。$ab$的值是多少？

(A) $\frac{1}{5}$ $\frac{1}{5}$

(B) $\frac{2}{5}$ $\frac{2}{5}$

(D) 1 1

(E) 4 4

Answer

Correct choice: (C)

正确答案：（C）

Solution

The total area of the four congruent triangles formed by the squares is $5-4 = 1$. Therefore, the area of one of these triangles is $\frac{1}{4}$. The height of one of these triangles is $a$ and the base is $b$. Using the formula for area of the triangle, we have $\frac{ab}{2} = \frac{1}{4}$. Multiply by $2$ on both sides to find that the value of $ab$ is $\boxed{\textbf{(C)}\ \frac{1}2}$.

两个正方形形成的四个全等三角形的总面积为$5-4 = 1$。因此，一个三角形的面积为$\frac{1}{4}$。该三角形的高为$a$，底为$b$。利用三角形面积公式，$\frac{ab}{2} = \frac{1}{4}$。两边乘以$2$，得$ab = \boxed{\textbf{(C)}\ \frac{1}2}$。

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