1981 AMC 12 Problem 11

Attempt Problem 11 of the 1981 AMC 12 below, then check your answer against the professionally curated solution from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 1981 AMC 12 solutions, or check the answer key.

All problems are used with official legal permission of the Mathematical Association of America (MAA).

11.

The three sides of a right triangle have integral lengths which form an arithmetic progression. One of the sides could have length

22

58

81

91

361

Represent the three equally spaced sides as

s-d,

s,

s+d

Every such integral right triangle is a multiple of a

3

4

5

triangle

If the sides are

s-d,

s,

s+d,

the Pythagorean equation gives

(s-d)^2+s^2=(s+d)^2,

hence

s=4k

and

d=k

for some positive integer

k.

Thus the sides are

3k,

4k,

5k.

Of the choices,

81=3\cdot27

can occur.

Therefore, the correct answer is C.

Problem 11 in other years