AMC10 2013 A

AMC10 2013 A · Q12

AMC10 2013 A · Q12. It mainly tests Triangles (properties), Similarity.

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 $DE$ and $EF$ are parallel to $\overline{AC}$ and $\overline{AB}$, respectively. What is the perimeter of parallelogram $ADEF$?

在$\triangle ABC$中，$AB = AC = 28$且$BC = 20$。点$D, E, F$分别在边$\overline{AB}, \overline{BC},$和$\overline{AC}$上，使得$DE$平行于$\overline{AC}$，$EF$平行于$\overline{AB}$。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

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

因为$EF \parallel AB$，所以$\triangle FEC$与$\triangle ABC$相似，且$FE = FC$。因此$ADEF$周长的一半是$AF + FE = AF + FC = AC = 28$。整个周长是56。

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