Question

点 $P$ 到某圆圆心的距离为 $9$ ，该圆的半径为 $15$ 。经过 $P$ 且长度为整数的不同弦共有多少条？

Point $P$ is $9$ units from the center of a circle of radius $15.$ How many different chords of the circle contain $P$ and have integer lengths?

$11$

$12$

$13$

$14$

$29$

答案：B

知识点：circle chord distance counting

难度评级：1610

小提示：

求经过 $P$ 的最短弦和最长弦

Find the shortest and longest chord through $P$

大提示：

除两个极值长度外，每个可取的弦长都对应两个方向

Except at the extreme lengths, each attainable chord length occurs in two directions

解答：

经过 $P$ 的最长弦是长为 $30$ 的直径。最短弦垂直于经过 $P$ 的半径，其长度为 $2\sqrt{15^2-9^2}=24\text{。}$ 当弦在这两个位置之间转动时，其长度连续地从 $24$ 变到 $30$ 。两个端点长度各出现一次，而中间的五个整数长度 $25,26,27,28,29$ 各出现两次。因此总数为 $1+2(5)+1=12$ 。

因此正确答案为 B。

The longest chord through $P$ is a diameter of length $30.$ The shortest is perpendicular to the radius through $P,$ and has length $2\sqrt{15^2-9^2}=24.$ As the chord rotates between these positions, its length varies continuously from $24$ to $30.$ The endpoint lengths each occur once, while each of the five interior integer lengths $25,26,27,28,29$ occurs twice. Thus the count is $1+2(5)+1=12.$

Thus the correct answer is B.

1991 AMC 12 第 10 题

答案：B

难度评级：1610

小提示：

大提示：

解答：

其他年份的第 10 题