In the ABC triangle, the AC side is equal to 21; the BH height is 12; the sine of the angle A

In the ABC triangle, the AC side is equal to 21; the BH height is 12; the sine of the angle A is equal to 0.6. Find the length of the segment CH.

Consider a triangle ABC, AC = 21, sin A = 0.6. The height BH is perpendicular to AC, and divides the triangle into two rectangular ones ACH and ABH. AC = AH + CH.
Consider a triangle ABH, <H = 90 °, AB – hypotenuse, AH and BH – legs.
We write the expression for determining the sine of angle A, as the ratio of the opposite leg to the hypotenuse:
sin A = BH / AB.
From here we find AB:
AB = BH / sin A = 12 / 0.6 = 20.
Angle A corresponding to sine <A = 37 °.
We write the expression for determining the cosine of angle A, as the ratio of the adjacent leg to the hypotenuse:
cos A = AH / AB
From here we find AH:
AH = cos A * AB = 0.79 * 20 = 15.8.
Determine the length of the segment CH:
AC = AH + CH
CH = AC-AH = 21-15.8 = 5.2.
Answer: the length of the segment CH = 5.2.