1996 AMC 12 Problem 15

Question

Two opposite sides of a rectangle are each divided into

n

congruent segments, and the endpoints of one segment are joined to the center to form triangle

A.

The other sides are each divided into

m

congruent segments, and the endpoints of one of these segments are joined to the center to form triangle

B.

[See figure for

n=5,

m=7.

] What is the ratio of the area of triangle

A

to the area of triangle

B?

1

\frac mn

\frac nm

\frac{2m}{n}

\frac{2n}{m}

Write the rectangle’s side lengths as

w

and

h

Each triangle has a base that is one divided segment and an altitude equal to half the opposite side length

Triangle

A

has base

\frac{w}{n}

and altitude

\frac{h}{2},

so its area is

\frac{wh}{4n}.

Triangle

B

has base

\frac{h}{m}

and altitude

\frac{w}{2},

so its area is

\frac{wh}{4m}.

Their ratio is

\frac{m}{n},

and the correct answer is B.

Problem 15 in other years