AMC12 2009 A

AMC12 2009 A · Q10

AMC12 2009 A · Q10. It mainly tests Triangles (properties), Inequalities with integers (floor/ceiling basics).

In quadrilateral $ABCD$, $AB = 5$, $BC = 17$, $CD = 5$, $DA = 9$, and $BD$ is an integer. What is $BD$?

在四边形 $ABCD$ 中，$AB = 5$，$BC = 17$，$CD = 5$，$DA = 9$，且 $BD$ 是整数。$BD$ 是多少？

(A) 11 11

(B) 12 12

(D) 14 14

(E) 15 15

Answer

Correct choice: (C)

正确答案：（C）

Solution

By the triangle inequality we have $BD < DA + AB = 9 + 5 = 14$, and also $BD + CD > BC$, hence $BD > BC - CD = 17 - 5 = 12$. We get that $12 < BD < 14$, and as we know that $BD$ is an integer, we must have $BD=\boxed{13}$.

由三角不等式可得 $BD < DA + AB = 9 + 5 = 14$，并且 $BD + CD > BC$，因此 $BD > BC - CD = 17 - 5 = 12$。所以 $12 < BD < 14$，又因为 $BD$ 是整数，必有 $BD=\boxed{13}$。

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