One of the legs of a right-angled triangle is 15. And the projection of the second leg onto the hypotenuse

One of the legs of a right-angled triangle is 15. And the projection of the second leg onto the hypotenuse is 16. What is the diameter of the circle circumscribed about this triangle?

1. The first leg of a right-angled triangle is: A = 15 cm;

2. The second leg is equal to: B cm;

3. Its projection onto the hypotenuse is: C2 = 16 cm;

4. The second part of the hypotenuse is equal to: C1 cm;

5. The total length of the hypotenuse: C = (C1 + C2) cm;

6. Length of the perpendicular to the hypotenuse: D cm;

7. By the Pythagorean theorem, we compose the equations:

C1² = A² – D²;

C2² = B² – D²;

A² – C1² = B² – C2²

A² – (C – C2) ² = (C² – A²) – C2²;

A² – (C² – 2 * C * C2 + C2²) = C² – A² – C2²

2 * C² – 2 * C2 * C – 2 * A² = 0;

C² – 16 * C – 15² = 0;

C1,2 = 8 + – sqrt (8² + 225) = 8 + – 17;

A negative root is meaningless;

C = 8 + 17 = 25 cm.

Answer: the diameter of the circumscribed circle is 25 centimeters.