In a right-angled triangle ABC, the leg AC = 90, and the height CH, dropped to the hypotenuse = 72

In a right-angled triangle ABC, the leg AC = 90, and the height CH, dropped to the hypotenuse = 72 Find the sine of the angle ABC.

In the right-angled triangle AСН, we find the leg AН (according to the Pythagorean theorem):
AH = √ (AC² – CH²) = √ (8100 – 5184) = √2916 = 54
Let us use the property of the height drawn to the hypotenuse from the right angle and find BH.
CH² = AH * BH,
BH = CH² / AH = 5184/54 = 96.
Let’s find the hypotenuse AB of the triangle ABC, and then the sine of the angle B.
AB = AH + BH = 54 + 96 = 150
Sin B = AC / AB = 90/150 = 0.6

You can find the sine of angle B through the cosine of angle A.
After finding AH, we find the cosine of the angle A in the triangle AСН:
cos A = AH / AC = 54/90 = 0.6.
Cos A = sin B = 0.6.
Answer: The sine of the angle ABC is 0.6.