AMC8 2007

AMC8 2007 · Q14

AMC8 2007 · Q14. It mainly tests Triangles (properties), Pythagorean theorem.

The base of isosceles △ABC is 24 and its area is 60. What is the length of one of the congruent sides?

等腰三角形 △ABC 的底边长为 24，其面积为 60。求一对全等边的长度。

(A) 5 5

(B) 8 8

(D) 14 14

(E) 18 18

Answer

Correct choice: (C)

正确答案：（C）

Solution

(C) Let $BD$ be the altitude from $B$ to $AC$ in $\triangle ABC$. Then $60=$ the area of $\triangle ABC=\frac{1}{2}\cdot 24\cdot BD$, so $BD=5$. Because $\triangle ABC$ is isosceles, $\triangle ABD$ and $\triangle CBD$ are congruent right triangles. This means that $AD=DC=\frac{24}{2}=12$. Applying the Pythagorean Theorem to $\triangle ABD$ gives $$ AB^2=5^2+12^2=169=13^2, $$ so $AB=13$.

（C）设$BD$为$\triangle ABC$中从$B$到$AC$的高。则$60=$ $\triangle ABC$的面积$=\frac{1}{2}\cdot 24\cdot BD$，所以$BD=5$。因为$\triangle ABC$是等腰三角形，$\triangle ABD$与$\triangle CBD$是全等的直角三角形。这意味着$AD=DC=\frac{24}{2}=12$。对$\triangle ABD$应用勾股定理得 $$ AB^2=5^2+12^2=169=13^2, $$ 所以$AB=13$。

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