central angles, chords, proving

Question

Rex assigned distinct points A, B, C, and D on a circle with diameter 25. He connects the consecutive points constructing quadrilateral ABCD. If angle A is equal to 90 degrees, and all sides are distinct integers, what is the perimeter of quadrilateral ABCD?

Answer 1

In acyclic quadrilateral opposite angles are supplementary.

Therefore the quadrilateral is made up of 2 right triangles with common hypotenuse which is a diameter=25.

The only right triangle with integer sides and diameter 25 has sides of 15 and 20 (a so-called 3-4-5 right triangle).