The bases of the trapezoid are 20 cm and 12 cm. The center of the circumscribed circle

The bases of the trapezoid are 20 cm and 12 cm. The center of the circumscribed circle lies on the larger base. Find the area of the trapezoid.

Let’s start by remembering that only an isosceles trapezoid can be inscribed into a circle.

From the condition it is known that the center of the circle is located on a larger base, which means that the angle formed by the lateral side and the diagonal is 90 °, since it rests on the diameter.

Let us introduce the notation for the trapezoid ABCD.

Then BC = 12, AD = 20.

BH is the height.

Let’s apply the height property of a right triangle.

The height of a right-angled triangle, lowered from an obtuse angle, divides the base into segments, the smaller of which is equal to the half of the difference between the bases, the larger is their half-sum.

AH = (AD – BC): 2 = (20 – 12): 2 = 4;

DH = (AD + BC): 2 = 16.

Let’s apply the property of the height of a right-angled triangle: the height drawn to the hypotenuse is the average proportional between the projections of the legs onto the hypotenuse.

BH ^ 2 = AH * DH;

BH = √4 * 16 = 8;

The area of ​​the trapezoid is equal to the product of the height and the half-sum of the bases.

S = 8 * 16 = 128 sq. units.