An isosceles trapezoid is described around the circle, the lengths of the bases of which are equal to 3

| April 29, 2021 | education

An isosceles trapezoid is described around the circle, the lengths of the bases of which are equal to 3 and 6. Find the radius of the circle.

If a circle can be inscribed into a trapezoid, then the sum of the lengths of its bases is equal to the sum of the lengths of its lateral sides.

BC + AD = AB + CD = 3 + 6 = 9 cm.

Since the trapezoid is isosceles, BC = AD = 9/2 = 4.5 cm.

The height of the trapezoid BH is equal to the diameter of the inscribed circle and divides the larger base into two segments, the length of the smaller of which is equal to the half-difference of the bases.

AH = (AD – BC) / 2 = (6 – 3) / 2 = 1.5 cm.

In a right-angled triangle ABN, according to the Pythagorean theorem, BH ^ 2 = AB ^ 2 – AH ^ 2 = 20.25 – 2.25 = 18.

BH = 3 * √2 cm.

Then R = BH / 2 = 3 * √2 / 2 = 1.5 * √2 cm.

Answer: The radius of the circle is 1.5 * √2 cm.



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