# An isosceles trapezoid is described near the circle, the bases of which are 6 centimeters

**An isosceles trapezoid is described near the circle, the bases of which are 6 centimeters and 24 centimeters. Find the radius of the circle and the area of the trapezoid**

Since the trapezoid is described around a circle, the sums of the lengths of its opposite sides are equal.

AC + CD = BC + AD = 6 + 24 = 30 cm.

Since the trapezoid is isosceles, then AB = CD = 30/2 = 15 cm.

Since the trapezoid is isosceles, the height ВН divides the base into two segments, the length of the smaller of which is equal to the half-difference of the lengths of the bases of the trapezoid.

AH = (AD – BC) / 2 = (24 – 6) / 2 = 9 cm.

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

BH ^ 2 = AB ^ 2 – AH ^ 2 = 225 – 81 = 144.

BH = 12 cm.

The radius of the circle is half the height of the trapezoid. R = BH / 2 = 12/2 = 6 cm.

The area of the trapezoid is equal to: Savsd = (ВС + АD) * ВН / 2 = (6 + 24) * 12/2 = 180 cm2.

Answer: The radius of the circle is 6 cm, the area is 180 cm2.