AMC8 2011

AMC8 2011 · Q20

AMC8 2011 · Q20. It mainly tests Pythagorean theorem, Area & perimeter.

Quadrilateral ABCD is a trapezoid, AD = 15, AB = 50, BC = 20, and the altitude is 12. What is the area of the trapezoid?

四边形ABCD是一个梯形，AD = 15，AB = 50，BC = 20，高为12。梯形的面积是多少？

(A) 600 600

(B) 650 650

(D) 750 750

(E) 800 800

Answer

Correct choice: (D)

正确答案：（D）

Solution

If you draw altitudes from $A$ and $B$ to $CD,$ the trapezoid will be divided into two right triangles and a rectangle. You can find the values of $a$ and $b$ with the Pythagorean theorem. \[a=\sqrt{15^2-12^2}=\sqrt{81}=9\] \[b=\sqrt{20^2-12^2}=\sqrt{256}=16\] $ABYX$ is a rectangle so $XY=AB=50.$ \[CD=a+XY+b=9+50+16=75\] The area of the trapezoid is \[12\cdot \frac{(50+75)}{2} = 6(125) = \boxed{\textbf{(D)}\ 750}\]

如果从$A$和$B$向CD作高线，梯形将被分成两个直角三角形和一个矩形。你可以用勾股定理求出$a$和$b$的值。 \[a=\sqrt{15^2-12^2}=\sqrt{81}=9\] \[b=\sqrt{20^2-12^2}=\sqrt{256}=16\] $ABYX$是一个矩形，所以$XY=AB=50$。 \[CD=a+XY+b=9+50+16=75\] 梯形的面积为 \[12\cdot \frac{(50+75)}{2} = 6(125) = \boxed{\textbf{(D)}\ 750}\]

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