A circle is inscribed in an isosceles trapezoid, one of the angles of which is 30 °, and the area is 72.

| September 23, 2021 | education

A circle is inscribed in an isosceles trapezoid, one of the angles of which is 30 °, and the area is 72. Find the radius of this circle.

We draw from the top of the angle B to the height of BH.

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

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

We use the formula for the area of a trapezoid.

Savsd = (ВС + АD) * BН / 2.

Then Savsd = 2 * AB * BН / 2 = AB * BН.

In a right-angled triangle ABH, the BH leg lies opposite the angle 30, then BH = AB / 2, and AB = 2 * BH. Substitute in the area equation.

Savsd = 2 * BH * BH = 2 * BH2 = 72.

BH2 = 72/2 = 36.

BH = 6 cm.

Then the radius of the circle is: R = BH / 2 = 6/2 = 3 cm.

Answer: The radius of the circle is 3 cm.



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