In triangle ABC, angle C is 90 ‘, CH is height, AB = 27, sin A = 2/3. Find BH.

In triangle ABC it is known:

Angle C is 90 °;
Height CH;
AB = 27;
sin a = 2/3.
Find BH.

1) If sin a and AB are known, then we can find BC.

sin a = BC / AB (ratio of the opposite leg to the hypotenuse);

Hence, BC = AB * sin a;

Substitute the known values.

BC = 27 * 2/3 = 27/3 * 2 = 9 * 2 = 18;

2) Since, sin a = cos b, then cos b = 2/3;

3) Consider a triangle CHB, where the angle H = 90 °.

Find BH.

cos b = BH / BC;

BH = BC * cos b;

BH = 18 * 2/3 = 18/3 * 2 = 6 * 3 = 12;

Hence, BH = 12.

Answer: BH = 12.