In an acute-angled triangle ABC, the height AH is 5√91. Side AB is 50. Find the cos of angle B

In a right-angled triangle, the cosine of the angle is the ratio of the adjacent leg to the hypotenuse. Thus, for our triangle, the following equality will be true:

cos B = BH / AB;

Let’s use the Pythagorean theorem and find the adjacent leg BH:

AB ^ 2 = BH ^ 2 + AH ^ 2;

BH ^ 2 = AB ^ 2 – AH ^ 2;

BH = √ (AB ^ 2 – AH ^ 2);

Substitute the values and find:

BH = √ (50 ^ 2 – (5 * √91) ^ 2) = √ (2500 – 25 * 91) = √ (25 * 100 – 25 * 91) = √ (25 * (100 – 91)) = 5 * √9 = 5 * 3 = 15.

Thus:

cos B = BH / AB = 15/50 = 3/10 = 0.3.

Answer: 0.3.