AMC8 2017

AMC8 2017 · Q16

AMC8 2017 · Q16. It mainly tests Area & perimeter, Ratios in geometry.

In the figure shown below, choose point D on side BC so that $\triangle ACD$ and $\triangle ABD$ have equal perimeters. What is the area of $\triangle ABD$?

如下图所示，在边BC上选择点D，使得$ riangle ACD$和$ riangle ABD$的周长相等。$ riangle ABD$的面积是多少？

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

(B) $\frac{3}{2}$ $\frac{3}{2}$

(D) $\frac{12}{5}$ $\frac{12}{5}$

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

Answer

Correct choice: (D)

正确答案：（D）

Solution

Answer (D): Because the perimeters of $\triangle ADC$ and $\triangle ADB$ are equal, $CD = 3$ and $BD = 2$. $\triangle ADC$ and $\triangle ADB$ have the same altitude from $A$, so the area of $\triangle ADC$ will be $\frac{3}{5}$ of the area of $\triangle ABC$, and $\triangle ADB$ will be $\frac{2}{5}$ of the area of $\triangle ABC$. The area of $\triangle ABC$ is $\frac{1}{2}\cdot 3\cdot 4 = 6$, so the area of $\triangle ADB$ is $\frac{2}{5}\cdot 6 = \frac{12}{5}$.

答案 (D)：由于 $\triangle ADC$ 和 $\triangle ADB$ 的周长相等，$CD=3$ 且 $BD=2$。 $\triangle ADC$ 和 $\triangle ADB$ 从 $A$ 作的高相同，所以 $\triangle ADC$ 的面积是 $\triangle ABC$ 面积的 $\frac{3}{5}$，而 $\triangle ADB$ 的面积是 $\triangle ABC$ 面积的 $\frac{2}{5}$。$\triangle ABC$ 的面积为 $\frac{1}{2}\cdot 3\cdot 4=6$，因此 $\triangle ADB$ 的面积为 $\frac{2}{5}\cdot 6=\frac{12}{5}$。

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