Question

等腰三角形 $\triangle ABC$ 的底边长为 $24$ ，面积为 $60$ 。其中一条全等边的长度是多少？

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

$5$

$8$

$13$

$14$

$18$

从顶点向底边作高

Draw the altitude from the top vertex to the base.

在等腰三角形中，这条高会把底边平分

In an isosceles triangle, that altitude splits the base into two equal parts.

作 $\overline{BD}$ 为从 $B$ 到 $\overline{AC}$ 的高。

则 $60 = \dfrac{1}{2} \cdot BD \cdot 24\text{，}$ 得到 $BD = 5$ 。

在 $\triangle ABD$ 中使用勾股定理： $AB^2 = 5^2 + 12^2 = 169 = 13^2\text{。}$ 因此 $AB = 13$ 。

所以正确答案是 C。

Construct $\overline{BD}$ as the altitude from $B$ to $\overline{AC}.$

Then $60 = \dfrac{1}{2} \cdot BD \cdot 24,$ which gives us that $BD = 5.$

From this, we apply the Pythagorean Theorem on $\triangle ABD:$ $AB^2 = 5^2 + 12^2 = 169 = 13^2.$ This gives us that $AB = 13.$

Thus, C is the correct answer.