The center of the circle circumscribed about the trapezoid lies on its larger base.

univerkov | June 11, 2021 | education

The center of the circle circumscribed about the trapezoid lies on its larger base. the lateral side of the trapezoid is 15, the radius of the circle is 12.5. find the area of the trapezoid.

Since the center of the circle lies on the larger base of the trapezoid, this base is the diameter of the circumscribed circle. AD = 2 * R = 2 * 12.5 = 25 cm.

Let’s draw a diagonal BD. The inscribed angle ABD is based on the diameter of the circle, then triangle ABD is rectangular.

By the Pythagorean theorem, BD ^ 2 = AD ^ 2 – AB ^ 2 = 625 – 225 = 400.BD = 20 cm.

Determine the area of the triangle ABD. Savd = AB * BD / 2 = 15 * 20/2 = 150 cm2.

Also Savd = AD * ВН / 2.

ВН = 2 * Savd / AD = 2 * 150/25 = 12 cm.

In a right-angled triangle ABН, according to the Pythagorean theorem, we determine the length of the leg AН.

AH ^ 2 = AB ^ 2 – BH ^ 2 = 225 – 144 = 81.

AH = 9 cm.

Since the trapezoid is inscribed in a circle, it is isosceles, then AH = (AD – BC) / 2.

BC = AD – 2 * AH = 25 – 18 = 7 cm.

Determine the area of the trapezoid.

Savsd = (ВС + АD) * ВН / 2 = (7 + 25) * 12/2 = 192 cm2.

Answer: The area of the trapezoid is 192 cm2.

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