AMC12 2009 B

AMC12 2009 B · Q4

AMC12 2009 B · Q4. It mainly tests Triangles (properties), Area & perimeter.

A rectangular yard contains two flower beds in the shape of congruent isosceles right triangles. The remainder of the yard has a trapezoidal shape, as shown. The parallel sides of the trapezoid have lengths $15$ and $25$ meters. What fraction of the yard is occupied by the flower beds?

一个矩形院子里有两个形状为全等的等腰直角三角形的花坛。院子剩余部分呈梯形，如图所示。梯形的两条平行边长分别为 $15$ 米和 $25$ 米。花坛占院子面积的几分之几？

(A) \frac{1}{8} \frac{1}{8}

(B) \frac{1}{6} \frac{1}{6}

(D) \frac{1}{4} \frac{1}{4}

(E) \frac{1}{3} \frac{1}{3}

Answer

Correct choice: (C)

正确答案：（C）

Solution

Each triangle has leg length $\frac 12 \cdot (25 - 15) = 5$ meters and area $\frac 12 \cdot 5^2 = \frac {25}{2}$ square meters. Thus the flower beds have a total area of $25$ square meters. The entire yard has length $25$ m and width $5$ m, so its area is $125$ square meters. The fraction of the yard occupied by the flower beds is $\frac {25}{125} = \boxed{\frac15}$. The answer is $\mathrm{(C)}$.

每个三角形的直角边长为 $\frac 12 \cdot (25 - 15) = 5$ 米，面积为 $\frac 12 \cdot 5^2 = \frac {25}{2}$ 平方米。因此花坛的总面积为 $25$ 平方米。整个院子长 $25$ 米、宽 $5$ 米，所以面积为 $125$ 平方米。花坛占院子的比例为 $\frac {25}{125} = \boxed{\frac15}$。答案是 $\mathrm{(C)}$。

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