In triangle ABC, angle C is 90 °, AB = 4 √15, cosA = 0.25. Find the height CH.

In the ABC triangle it is known:

Angle C = 90 °;
AB = 4√15;
cos A = 0.25.
Find the height CH.

Decision:

1) cos a = AC / AB (the ratio of the adjacent leg to the hypotenuse);

From here we express AC.

AC = AB * cos A = 4√15 * 0.25 = 4√15 * 1/4 = 4/4 * √15 = √15;

2) Consider a right-angled triangle ACН with a right angle H.

sin a = √ (1 – cos ^ 2 a) = √ (1 – (3/4) ^ 2) = √ (1 – 9/16) = √7 / √16 = √7 / 4;

sin a = CH / AC;

CH = AC * sin a;

Substitute the known values and then find the height.

CH = √15 * √7 / 4 = √105 / 4.

Answer: CH = √105 / 4.