The diagonal of an isosceles trapezoid is perpendicular to the lateral side. find the area of the trapezoid

The diagonal of an isosceles trapezoid is perpendicular to the lateral side. find the area of the trapezoid, if its height is 12cm, diagonal is 20cm.

By the property of an isosceles trapezoid, the height drawn from the apex of an obtuse angle to the larger base divides it into two segments (AH and DH), the larger of which is equal to the half-sum of the bases, and the smaller one is the half-difference of the bases.

AH = (AD + BC) / 2.

Consider a right-angled triangle ACH and, by the Pythagorean theorem, define the leg AH.

AH ^ 2 = AC ^ 2 – CH ^ 2 = 20 ^ 2 – 12 ^ 2 = 400 – 144 = 256.

AH = 16 cm.

Determine the area of the trapezoid.

S = (AD + BC) / 2 * CH = AH * CH / 2 = 16 * 12 = 192 cm2.

Answer: The area of the trapezoid is 192 cm2.