AMC 8 2003 Problem 21

AMC8 2003 年第 21 题

四边形与多边形★★★☆☆

The area of trapezoid

ABCD

164\text{ cm}^2

. The altitude is 8 cm,

AB

is 10 cm, and

CD

is 17 cm. What is

BC

, in centimeters?

梯形 ABCD 的面积是 164 平方厘米，高为 8 厘米，AB 的长为 10 厘米，CD 的长为 17 厘米。则 BC 的长为多少厘米?

A.9
B.10
C.12
D.15
E.20

正确答案 · Correct Answer

解析 · Solution

Solution 1

Using the formula for the area of a trapezoid, we have

\displaystyle 164=8\left(\frac{BC+AD}{2}\right)

. Thus BC+AD=41. Drop perpendiculars from

B

AD

and from

C

AD

and let them hit

AD

E

and

F

respectively. Note that each of these perpendiculars has length 8. From the Pythagorean Theorem, AE=6 and DF=15 thus AD=BC+21. Substituting back into our original equation we have BC+BC+21=41 thus

BC=\text{(B)10}

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