An isosceles trapezoid is described near the circle, in which the point of contact cuts off a segment of 8 cm

| May 22, 2021 | education

An isosceles trapezoid is described near the circle, in which the point of contact cuts off a segment of 8 cm on the lateral side, counting from the top of the smaller base. Find the area of a trapezoid if the sum of its bases is 52 cm.

From the center of the circle, draw the radii OP and OK to the tangency points of the trapezoid and the circle.

By the property of a tangent drawn from one point, the lengths of the tangents are equal.

Point B is a common point, segments BK and BP are tangent, then BP = BK = 8 cm.

The radius of the OP, drawn to the base of the aircraft, divides it in half. Then BC = 2 * BP = 2 * 8 = 16 cm.

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

BC + AD = AB + CD = 2 * AB, since AB = CD.

52 = 2 * AB.

AB = 52/2 = 26 cm.

Then the segment AK = 26 – 8 = 18 cm.

Determine the radius of the circle.

OK2 = AK * BK = 18 * 8 = 144.

OK = R = 12 cm.

Then the height of the trapezoid is:

PH = 2 * R = 2 * 12 = 24 cm.

Determine the area of ​​the trapezoid.

S = (AD + BC) * PH / 2 = 52 * 24/2 = 624 cm2.

Answer: The area of ​​the trapezoid is 624 cm.



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