A circle is inscribed in an isosceles trapezoid. What is the acute angle of a trapezoid if the point of tangency

A circle is inscribed in an isosceles trapezoid. What is the acute angle of a trapezoid if the point of tangency divides the lateral side into segments, the length ratio of which is 3: 1?

Let the length of the segment BM = X cm, then the length of the segment AM = 3 * X cm.

By the property of a tangent drawn from one point, the segment BP = BM = X cm, the segment AM = AK = 3 * X cm.

Let’s draw the height BH, then the quadrangle BPKH is a rectangle, and HK = BP = X cm.

Then AH = AK – HK = 3 * X – X = 2 * X cm.

In a right-angled triangle ABH, we determine the value of the angle BAH.

AB = AM + BM = 3 * X + X = 4 * X.

CosBAH = AH / AB = 2 * X / 4 * X = 1/2.

Angle BAH = arcos (1/2) = 60.

Answer: The acute angle of the trapezoid is 60.