AMC10 2018 A

AMC10 2018 A · Q16

AMC10 2018 A · Q16. It mainly tests Pythagorean theorem, Coordinate geometry.

Right triangle ABC has leg lengths AB = 20 and BC = 21. Including AB and BC, how many line segments with integer length can be drawn from vertex B to a point on hypotenuse AC?

直角三角形 ABC 有直角边长 AB = 20 和 BC = 21。包括 AB 和 BC 在内，从顶点 B 到斜边 AC 上的点能画出多少条整数长度的线段？

(A) 5 5

(B) 8 8

(D) 13 13

(E) 15 15

Answer

Correct choice: (D)

正确答案：（D）

Solution

Answer (D): The area of $\triangle ABC$ is 210. Let $D$ be the foot of the altitude from $B$ to $AC$. By the Pythagorean Theorem, $AC=\sqrt{20^2+21^2}=29$, so $210=\frac12\cdot29\cdot BD$, and $BD=14\frac{14}{29}$. Two segments of every length from 15 through 19 can be constructed from $B$ to $AC$. In addition to these 10 segments and the 2 legs, there is a segment of length 20 from $B$ to a point on $AC$ near $C$, for a total of 13 segments with integer length.

答案（D）：$\triangle ABC$ 的面积是 210。设 $D$ 为从 $B$ 向 $AC$ 作高的垂足。由勾股定理，$AC=\sqrt{20^2+21^2}=29$，所以 $210=\frac12\cdot29\cdot BD$，从而 $BD=14\frac{14}{29}$。从 $B$ 到 $AC$ 可以构造出长度为 15 到 19（含）的每一种长度各两条线段。除这 10 条线段和两条直角边外，还存在一条从 $B$ 到 $AC$ 上靠近 $C$ 的点的长度为 20 的线段，因此整数长度的线段总数为 13 条。

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