AMC 8 2017 Problem 18

AMC8 2017 年第 18 题

面积与周长（综合）★★★☆☆

In the non-convex quadrilateral ABCD shown below,

\angle BCD

is a right angle, AB=12 , BC=4 , CD=3 , and AD=13 . What is the area of quadrilateral ABCD ?

在如下图所示的非凸四边形 ABCD 中， AD=13.那么四边形 ABCD 的面积是多少？

正确答案 · Correct Answer

解析 · Solution

Solution 1

\triangle BCD

is a 3-4-5 right triangle. So the area of

\triangle BCD

is 6. Then we can use Heron's formula to compute the area of

\triangle ABD

whose sides have lengths 5,12,and 13. The area of

\triangle ABD

\sqrt{s(s-5)(s-12)(s-13)}

, where s is the semi-perimeter of the triangle, that is s=(5+12+13)/2=15. Thus, the area of

\triangle ABD

is 30, so the area of

ABCD

is 30-6 = 24. -

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