Find the radius of a circle circumscribed about a trapezoid if it is known that the middle line of the trapezoid is 14 cm

| February 26, 2021 | education

Find the radius of a circle circumscribed about a trapezoid if it is known that the middle line of the trapezoid is 14 cm, the side of the trapezoid is cm, and one of the bases is the diameter of this circle.

Since the trapezoid is inscribed in a circle, it is isosceles.

Let the length of the base BC = X cm, base AD = Y cm. Let us construct the height BH, which divides the larger base into two segments, the larger of which is equal to the middle line of the trapezoid.

DН = КМ = 14 cm, then АН = (Y – 14) cm.

Triangle ABD is rectangular since the inscribed angle ABD is based on the diameter AD.

The BH height is drawn from the top of the right angle to the hypotenuse, then AB ^ 2 = AD * AH.

(4 * √2) ^ 2 = Y * (Y – 14).

32 = Y ^ 2 – 14 * Y.

Y ^ 2 – 14 * Y – 32 = 0.

Let’s solve the quadratic equation.

Y1 = AD = D = 16cm.

Y ^ 2 = -2. (Doesn’t fit because <0).

Then R = D / 2 = 16/2 = 8 cm.

Answer: The radius of the circle is 8 cm.



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