Question

一个女生营地距一条直路 $300$ 杆。一个男生营地位于这条路上，距女生营地 $500$ 杆。现要在路上建一座食堂，使它到两个营地的距离完全相同。食堂到每个营地的距离为：

A girls’ camp is located $300$ rods from a straight road. On this road, a boys’ camp is located $500$ rods from the girls’ camp. It is desired to build a canteen on the road which shall be exactly the same distance from each camp. The distance of the canteen from each of the camps is:

$400$ 杆

$400$ rods

$250$ 杆

$250$ rods

$87.5$ 杆

$87.5$ rods

$200$ 杆

$200$ rods

以上答案均不正确

none of these

答案：E

知识点：坐标几何 Pythagorean theorem equidistance

难度评级：1470

小提示：

垂直距离与两营地间的距离构成一个 $300$ - $400$ - $500$ 直角三角形

The perpendicular and camp-to-camp distances form a $300$ - $400$ - $500$ right triangle

大提示：

将两个营地设在 $(0,300)$ 和 $(400,0)$ ，食堂设在 $(t,0)$

Place the camps at $(0,300)$ and $(400,0)$ , and put the canteen at $(t,0)$

解答：

将女生营地到道路的垂足设为 $(0,0)$ 。两个营地可分别设为 $G=(0,300)$ 和 $B=(400,0)$ 。若食堂为 $C=(t,0)$ ，由到两营地距离相等可得 $t^2+300^2=(400-t)^2\text{，}$ 所以 $t=87.5$ 。公共距离为 $BC=400-87.5=312.5$ 杆，不在选项中。

因此，正确答案是 E。

Let the foot of the perpendicular from the girls’ camp be $(0,0).$ The camps can be placed at $G=(0,300)$ and $B=(400,0).$ If the canteen is $C=(t,0),$ equidistance gives $t^2+300^2=(400-t)^2,$ so $t=87.5.$ The common distance is $BC=400-87.5=312.5$ rods, which is not listed.

Thus, the correct answer is E.

1953 AMC 12 第 41 题

答案：E

难度评级：1470

小提示：

大提示：

解答：

其他年份的第 41 题