The trapezoid is inscribed in a circle of radius 6.5, its larger base is the diameter of the circle.

| August 15, 2021 | education

The trapezoid is inscribed in a circle of radius 6.5, its larger base is the diameter of the circle. Find the area of a trapezoid if its midline is 9.

Since the larger base of the trapezoid lies on the diameter of the circle, the length of the larger base of the trapezoid is equal to two radii of the circle. AD = 2 * R = 2 * 6.5 = 13 cm.

Through the length of the larger base and the midline of the trapezoid, we determine the length of the smaller base.

KM = (BC + AD) / 2.

BC = 2 * KM – AD = 2 * 9 – 13 = 5 cm.

Let’s draw the height of the trapezium CH. Since the trapezoid is inscribed in a circle, it is isosceles, and the segment DH is equal to the half-difference of the lengths of the bases of the trapezoid.

DН = (АD – ВС) / 2 = (13 – 5) / 2 = 4 cm.

Then the segment OH = AD – DH = 6.5 – 4 = 2.5 cm.

In a right-angled triangle OCH, the length of the hypotenuse OS is equal to the radius of the circle, then by the Pythagorean theorem: CH ^ 2 = OS ^ 2 – OH ^ 2 = 6.5 ^ 2 – 2.5 ^ 2 = 42.25 – 6.25 = 36.

CH = 6 cm.

Determine the area of ​​the trapezoid.

S = KM * CH = 9 * 6 = 54 cm2.

Answer: The area of ​​the trapezoid is 54 cm2.



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