In the triangle ABC AC = BC, AB = 24, cosA = 24/25. Find the height of the CH.

Let AC = BC = x.

By the cosine theorem, the BC side is equal to:

(BC) ^ 2 = (AB) ^ 2 + (AC) ^ 2 – 2 * AB * AC * cosA = (24) ^ 2 + (AC) ^ 2 – 2 * 24 * AC * 24/25.

(x) ^ 2 = (24) ^ 2 + (x) ^ 2 – 2 * 24 * x * 24/25.

(2 * (24) ^ 2 * x) / 25 = (24) ^ 2.

2 * x = 25.

x = 12.5.

The ABC triangle is isosceles, since AC = BC = x.

Hence, AH = AB / 2 = 24/2 = 12.

Consider a triangle HAC – rectangular, AC – hypotenuse.

Then, by the Pythagorean theorem, we find the height HC.

(HC) ^ 2 = (AC) ^ 2 – (AH) ^ 2 = (12.5) ^ 2 – (12) ^ 2 = 156.25 – 144 = 12.25.

Therefore, HC = 3.5.