AMC 12B 2024 (Problem 21)
The measures of the smallest angles of three different right triangles sum to . All three triangles have side lengths that are primitive Pythagorean triples. Two of them are and . What is the perimeter of the third triangle?
Using the complementary-angle identity,
Using the cosine addition formula,
By Hints 2 and 3,
Since the side lengths form a primitive Pythagorean triple and ,
By the Pythagorean theorem,
Therefore,
The perimeter of the third triangle is
Since the side lengths form a primitive Pythagorean triple and ,
By the Pythagorean theorem,
Therefore,
The perimeter of the third triangle is
(C) 154