Question

在下图中，在边 $BC$ 上选择点 $D$ ，使得 $\triangle ACD$ 和 $\triangle ABD$ 的周长相等。 $\triangle ABD$ 的面积是多少？

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 $\triangle ABD?$

$\dfrac{3}{4}$

$\dfrac{3}{2}$

$2$

$\dfrac{12}{5}$

$\dfrac{5}{2}$

答案：D

知识点：面积比周长

难度评级：1420

小提示：

用周长相等条件求 $BD$ 和 $CD$

Use the equal-perimeter condition to find $BD$ and $CD$ .

大提示：

一旦 $BC$ 被分割，两个三角形从 $A$ 引出的高相同，所以面积按底边长度比较

Once $BC$ is split, the two triangles have the same altitude from $A$ , so compare areas by base lengths.

视频讲解：

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文字解答：

要使两个三角形周长相等， $\overline{BC}$ 必须分成 $\overline{CD} = 3$ 和 $\overline{BD} = 2$ 。

$\triangle ACD$ 和 $\triangle ABD$ 有相同的高，所以面积与底边长度成比例。这说明 $\triangle ABD$ 的面积是 $\dfrac{2}{5}$ 倍的 $\triangle ABC$ 的面积，即 $\dfrac{2}{5} \cdot 3 \cdot \dfrac{4}{2} = \dfrac{12}{5}\text{。}$

所以正确答案是 D。

The only way to split $\overline{BC}$ into two parts such that the two triangles have the same perimeter is if $\overline{CD} = 3$ and $\overline{BD} = 2.$

$\triangle ACD$ and $\triangle ABD$ have the same altitudes, so their areas are proportional to their bases. This means that the area of $\triangle ABD$ is $\dfrac{2}{5}$ the area of $\triangle ABC,$ which is $\dfrac{2}{5} \cdot 3 \cdot \dfrac{4}{2} = \dfrac{12}{5}.$

Thus, D is the correct answer.

2017 AMC 8 第 16 题