AMC12 2013 A

AMC12 2013 A · Q9

AMC12 2013 A · Q9. It mainly tests Similarity, Ratios in geometry.

In $\triangle ABC$, $AB = AC = 28$ and $BC = 20$. Points $D, E, F$ are on sides $\overline{AB}, \overline{BC},$ and $\overline{AC}$, respectively, such that $\overline{DE}$ and $\overline{EF}$ are parallel to $\overline{AC}$ and $\overline{AB}$, respectively. What is the perimeter of parallelogram $ADEF$?

在 $\triangle ABC$ 中，$AB = AC = 28$，$BC = 20$。点 $D, E, F$ 分别在边 $\overline{AB}, \overline{BC}, \overline{AC}$ 上，使得 $\overline{DE} \parallel \overline{AC}$，$\overline{EF} \parallel \overline{AB}$。平行四边形 $ADEF$ 的周长是多少？

(A) 48 48

(B) 52 52

(D) 60 60

(E) 72 72

Answer

Correct choice: (C)

正确答案：（C）

Solution

Answer (C): Because $\overline{EF}$ is parallel to $\overline{AB}$, it follows that $\triangle FEC$ is similar to $\triangle ABC$ and $\overline{FE}=\overline{FC}$. Thus half of the perimeter of $ADEF$ is $\overline{AF}+\overline{FE}=\overline{AF}+\overline{FC}=\overline{AC}=28$. The entire perimeter is 56.

答案（C）：因为 $\overline{EF}$ 平行于 $\overline{AB}$，所以 $\triangle FEC$ 与 $\triangle ABC$ 相似，且 $\overline{FE}=\overline{FC}$。因此，四边形 $ADEF$ 的周长的一半为 $\overline{AF}+\overline{FE}=\overline{AF}+\overline{FC}=\overline{AC}=28$。所以整个周长为 56。

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