1967 AMC 12 Problem 15

Question

The difference in the areas of two similar triangles is

18

square feet, and the ratio of the larger area to the smaller is the square of an integer. The area of the smaller triangle, in square feet, is an integer, and one of its sides is

3

feet. The corresponding side of the larger triangle, in feet, is:

12

9

6\sqrt2

6

3\sqrt2

Answer: D

Concepts:similarity area ratio divisibility

Difficulty rating: 1850

Small hint:

Let the integer side-scale factor be

k

Big hint:

If the smaller area is

T,

then

T(k^2-1)=18

Solution:

Let the side-scale factor be the integer

k\gt1

and the smaller area be the integer

T.

Then

T(k^2-1)=18.

Thus

k^2-1

is a positive divisor of

18.

The only resulting square

k^2

is

4,

so

k=2.

The corresponding side is

3k=6.

Therefore, the correct answer is D.