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

| June 19, 2021 | education

Find the radius of the circle circumscribed about the trapezoid if it is known that the middle line of the trapezoid is 14 cm, the lateral side is 4√2, and one of the bases of the trapezoid is the diameter of the circumscribed circle.

Let us construct the BH height and BD diagonal.

Since the trapezoid is inscribed in circles, the ABCD trapezoid is isosceles. The height BH divides the base of AD into two segments, the largest of which is equal to the length of the midline. DН = КМ = 14 cm.

Triangle ABD is rectangular since the angle ABD is based on the diameter of the circle.

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

Let the length AH = X cm, then AD = 14 + X cm.

(4 * √2) ^ 2 = X * (14 + X).

X ^ 2 + 14 * X – 32 = 0.

Let’s solve the quadratic equation.

X = AH = 2 cm.

Then AD = 2 + 14 = 16 cm.

R = АD / 2 = 16/2 = 8 cm.

Answer: The radius of the circle is 8 cm.



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