Find the base AB of an isosceles triangle ABC if AC = 7, cos A = 0.125.

From the condition we know that triangle ABC is isosceles, and we also know that AC = 7, and cos A = 0.125. We need to find the base of the triangle AB.

Let’s draw the height CH. And since the triangle is isosceles, the height in it will also be the median, or rather, it will divide the base in half:

AH = HB = 1/2 * AB.

Consider the resulting right-angled triangle and write cos A based on this. The cosine of the angle of a right-angled triangle is equal to the ratio of the adjacent leg to the hypotenuse:

cos A = AH / AC;

Let’s express the leg:

AH = cos A * AC = 7 * 0.125 = 0.875.

We are looking for the base: AB = 2 * AH = 2 * 0.875 = 1.75.