1953 AMC 12 第 41 题

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：

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