In an isosceles triangle ABC with a base AC, the lateral side AB is 10, and cos A = 0.3√11.

In an isosceles triangle ABC with a base AC, the lateral side AB is 10, and cos A = 0.3√11. Find the height drawn to the base.

Let BH be the height drawn to the base of the speaker. From the right-angled triangle ABH (angle H of the straight line), we express the cosine of the angle BAH and substitute the known values there. We get: cos BAH = AH / AB, 0.3√11 = AH / 10. Let’s find the side of AH. AH = 10 * 0.3√11 = 3√11. In a triangle ABH, according to the Pythagorean theorem (the square of the hypotenuse is equal to the sum of the squares of the legs): AB ^ 2 = AH ^ 2 + BH ^ 2. Let us express and find BH: BH ^ 2 = AB ^ 2-AH ^ 2, BH ^ 2 = 10 ^ 2- (3√11) ^ 2 = 100-99 = 1. Since the sizes of the sides can only be positive, only a positive value is suitable for us, i.e. BH = 1.
Answer: the height is 1.