AMC 8 2017 Problem 16

AMC8 2017 年第 16 题

三角形★★★☆☆

In the figure below, choose point D on

\overline{BC}

so that

\triangle ACD

and

\triangle ABD

have equal perimeters. What is the area of

\triangle ABD

如下图所示，在

\overline{BC}

上取点

D

，使得

\triangle ACD

和

\triangle ABD

的周长相等。那么

\triangle ABD

的面积是多少？

A. $\displaystyle \frac{3}{4}$
B. $\displaystyle \frac{3}{2}$
C.2
D. $\displaystyle \frac{12}{5}$
E. $\displaystyle \frac{5}{2}$

正确答案 · Correct Answer

解析 · Solution

Solution 1

We know that the perimeters of the two small triangles are 3+CD+AD and 4+BD+AD. Setting both equal and using BD+CD = 5, we have BD = 2 and CD = 3. Now, we simply have to find the area of

\triangle ABD

. Since

\displaystyle \frac{BD}{CD} = \frac{2}{3}

, we must have

\displaystyle \frac{[ABD]}{[ACD]} = 2/3

. Combining this with the fact that

\displaystyle [ABC] = [ABD] + [ACD] = \frac{3\cdot4}{2} = 6

, we get

\displaystyle [ABD] = \frac{2}{5} [ABC] = \frac{2}{5} \cdot 6 = \frac{12}{5}

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