# Given a rectangular trapezoid, the larger base of which is 12 cm, and the radius of the inscribed

**Given a rectangular trapezoid, the larger base of which is 12 cm, and the radius of the inscribed circle is 3 cm. Find the area of the trapezoid.**

From the center of the O circle, draw the radii OK, OH and OM, to the points of tangency.

By the property of a tangent drawn from one point, BK = BM = AH = AM = OM = 3 cm.

Then the length of the segment DH = AD – AH = 12 – 3 = 9 cm.

From a right-angled triangle DOH tgODH = OH / DH = 3/9 = 1/3.

If a circle is inscribed in a trapezoid, then the triangles formed by the side and the center of the circle are rectangular. Then the triangle COD is rectangular, and then the angle is ODH = COK.

In a right-angled triangle SOC tgO = KC / KO.

1/3 = KC / 3.

KC = 1 cm.

Then BC = BK + KC = 3 + 1 = 4 cm.

Determine the area of the trapezoid.

Savsd = (ВС + АD) * КН / 2 = (4 + 12) * 6/2 = 48 cm2.

Answer: The area of the trapezoid is 48 cm2.