Question

Two triangles have integral side lengths, with all sides being less than 50. They are similar but not congruent and smaller triangle has two side lengths identical with the larger triangle.

What is the sum of the side lengths of the larger triangle?

Answer 1

the answer is 57
and logic is -
Let the smaller triangle have side lengths a, b, c with a < b < c, and the larger triangle have side lengths b, c, d with b < c < d.

a, b, c, d form a geometric progression, and (a +b) > c. So the ratio is less than 1.618. Since all the sides are integral, a = f^3, b = f^3r (where r is the ratio), c = f^3r^2, and d = f^3r^3.

Since d < 50, d is an integer cube > 2, d must be 27. The ratio r therefore must b 1.5 so that a = 8: b = 12, c = 18.

12 + 18 + 27 = 57