# A circle is inscribed in an isosceles trapezoid (i.e., all sides of the trapezoid touch the circle).

A circle is inscribed in an isosceles trapezoid (i.e., all sides of the trapezoid touch the circle). Find the radius of the circle if the bases of the trapezoid are 8cm and 16cm.

Since a circle is inscribed in the trapezoid, the sum of the lengths of the lateral sides of the trapezoid is equal to the sum of the lengths of its lateral sides.

(AB + CD) = (BC + AD) = 8 + 16 = 24 cm.

Then AB = CD = 24/2 = 12 cm.

Let’s draw the height of the HV trapezoid. Since the trapezoid is isosceles, the length of the segment AH is equal to the half-difference of the lengths of the bases of the trapezoid.

AH = (AD – BC) / 2 = (16 – 8) / 2 = 8/2 = 4 cm.

From the right-angled triangle ABN, we determine, according to the Pythagorean theorem, the length of the leg BN.

BH ^ 2 = AB ^ 2 – AH ^ 2 = 144 – 16 = 128.

BH = 8 * √2 cm.

The radius of the inscribed circle is equal to half the length of the height of the trapezoid.

R = ОМ = ВН / 2 = 8 * √2 / 2 = 4 * √2 cm.