In a right-angled triangle ABC, the leg AC = 24, and the height of the CH, lowered to the hypotenuse

In a right-angled triangle ABC, the leg AC = 24, and the height of the CH, lowered to the hypotenuse, is 6√15. Find the sine of the angle ABC.

1. Applying the Pythagorean theorem, we calculate the length of the segment АН, which is the leg of the right-angled triangle АСН:

AH ^ 2 = AC ^ 2 – CH ^ 2 = 24 ^ – (6√15) ^ 2 = 576 – 540 = 36;

AH = √36 = 6;

2. The height of the CH, drawn to the hypotenuse AB, is equal to √АH x BH;

CH ^ 2 = AH x BH;

BH = CH ^ 2 / AH = 540/6 = 90.

3. Calculate the length of the hypotenuse AB:

90 + 6 = 96;

4. The sine of the angle at the vertex B is equal to AC / AB = 24/96 = 1/4.

Answer: the sine of the angle at the vertex B is 1/4.