1983 AMC 12 Problem 24

Question

How many non-congruent right triangles are there such that the perimeter in cm and area in

\text{cm}^2

are numerically equal?

none

1

2

4

infinitely many

Start with any shape of right triangle and scale all its side lengths by

k

Under scaling, perimeter is multiplied by

k

while area is multiplied by

k^2

Start with any right triangle having perimeter

P

and area

K.

Scaling every side by

k=\frac{P}{K}

produces perimeter

kP

and area

k^2K,

which are equal. Infinitely many nonsimilar right-triangle shapes exist, and the resulting triangles are therefore noncongruent.

Therefore, the correct answer is E.

Problem 24 in other years