AMC12 2014 B

AMC12 2014 B · Q9

AMC12 2014 B · Q9. It mainly tests Triangles (properties), Pythagorean theorem.

Convex quadrilateral $ABCD$ has $AB = 3$, $BC = 4$, $CD = 13$, $AD = 12$, and $\angle ABC = 90^\circ$, as shown. What is the area of the quadrilateral?

凸四边形 $ABCD$ 有 $AB = 3$，$BC = 4$，$CD = 13$，$AD = 12$，且 $\angle ABC = 90^\circ$，如图所示。四边形的面积是多少？

(A) 30 30

(B) 36 36

(D) 48 48

(E) 58.5 58.5

Answer

Correct choice: (B)

正确答案：（B）

Solution

Answer (B): By the Pythagorean Theorem, $AC = 5$. Because $5^2 + 12^2 = 13^2$, the converse of the Pythagorean Theorem applied to $\triangle DAC$ implies that $\angle DAC = 90^\circ$. The area of $\triangle ABC$ is $6$ and the area of $\triangle DAC$ is $30$. Thus the area of the quadrilateral is $6 + 30 = 36$.

答案（B）：由勾股定理，$AC = 5$。因为 $5^2 + 12^2 = 13^2$，将勾股定理的逆定理应用于 $\triangle DAC$ 可得 $\angle DAC = 90^\circ$。$\triangle ABC$ 的面积为 $6$，$\triangle DAC$ 的面积为 $30$。因此该四边形的面积为 $6 + 30 = 36$。

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