Question

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:

$400$ rods

$250$ rods

$87.5$ rods

$200$ rods

none of these

Answer: E

Concepts:coordinate geometry Pythagorean theorem equidistance

Difficulty rating: 1470

Small Hint:

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

Big Hint:

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

Solution:

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 Problem 41

Answer: E

Difficulty rating: 1470

Small Hint:

Big Hint:

Solution:

Problem 41 in Other Years