AMC10 2013 A

AMC10 2013 A · Q15

AMC10 2013 A · Q15. It mainly tests Systems of equations, Area & perimeter.

Two sides of a triangle have lengths 10 and 15. The length of the altitude to the third side is the average of the lengths of the altitudes to the two given sides. How long is the third side?

一个三角形的两边长分别为10和15。到第三边的垂线长度是到这两边垂线长度平均值。第三边有多长？

(A) 6 6

(B) 8 8

(D) 12 12

(E) 18 18

Answer

Correct choice: (D)

正确答案：（D）

Solution

Answer (D): Denote the length of the third side as $x$, and the altitudes to the sides of lengths 10 and 15 as $m$ and $n$, respectively. Then twice the area of the triangle is $10m = 15n = \frac{1}{2}x(m+n)$. This implies that $m=\frac{3}{2}n$, so $$15n=\frac{1}{2}x\left(\frac{3}{2}n+n\right)=\frac{5}{4}xn.$$ Therefore $15=\frac{5}{4}x$, and $x=12$.

答案（D）：设第三边的长度为 $x$，分别作到长度为 10 和 15 的边的高为 $m$ 和 $n$。则三角形面积的两倍为 $10m=15n=\frac{1}{2}x(m+n)$。由此可得 $m=\frac{3}{2}n$，所以 $$15n=\frac{1}{2}x\left(\frac{3}{2}n+n\right)=\frac{5}{4}xn.$$ 因此 $15=\frac{5}{4}x$，从而 $x=12$。

Topics