In an isosceles triangle ABCD with a base AC, the side = 10, and the cos of the angle A = 4 fifths.

In an isosceles triangle ABCD with a base AC, the side = 10, and the cos of the angle A = 4 fifths. Find the height drawn to the base.

Since, by condition, BH is the height of the ABC triangle, then the ABH triangle is rectangular.

In a right-angled triangle, the cosine of an acute angle is equal to the ratio of the length of the adjacent leg to the length of the hypotenuse.

CosA = AH / AB.

AH = AB * CosA = 10 * 4/5 = 8 cm.

In a right-angled triangle ABH, according to the Pythagorean theorem, we determine the length of the leg BH.

BH ^ 2 = AB ^ 2 – AH ^ 2 = 100 – 64 = 36.

BH = 6 cm.

Answer: The length of the height is 6 cm.