AMC12 2015 A

AMC12 2015 A · Q8

AMC12 2015 A · Q8. It mainly tests Pythagorean theorem, Area & perimeter.

The ratio of the length to the width of a rectangle is $4:3$. If the rectangle has diagonal of length $d$, then the area may be expressed as $kd^2$ for some constant $k$. What is $k$?

一个矩形的长宽比为 $4:3$。如果矩形的对角线长度为 $d$，则面积可表示为 $kd^2$，其中 $k$ 为某个常数。$k$ 是多少？

(A) $\frac{2}{7}$ $\frac{2}{7}$

(B) $\frac{3}{7}$ $\frac{3}{7}$

(D) $\frac{16}{25}$ $\frac{16}{25}$

(E) $\frac{3}{4}$ $\frac{3}{4}$

Answer

Correct choice: (C)

正确答案：（C）

Solution

Answer (C): Let the sides of the rectangle have lengths $3a$ and $4a$. By the Pythagorean Theorem, the diagonal has length $5a$. Because $5a=d$, the side lengths are $\frac{3}{5}d$ and $\frac{4}{5}d$. Therefore the area is $\frac{3}{5}d\cdot\frac{4}{5}d=\frac{12}{25}d^2$, so $k=\frac{12}{25}$.

答案（C）：设矩形的两边长分别为 $3a$ 和 $4a$。由勾股定理可知，对角线长为 $5a$。因为 $5a=d$，所以两边长分别为 $\frac{3}{5}d$ 和 $\frac{4}{5}d$。因此面积为 $\frac{3}{5}d\cdot\frac{4}{5}d=\frac{12}{25}d^2$，所以 $k=\frac{12}{25}$。

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