In a rectangular trapezoid, the acute angle is 60 degrees, and the distance from the center

| January 4, 2021 | education

In a rectangular trapezoid, the acute angle is 60 degrees, and the distance from the center of the inscribed circle to the obtuse angle is 10 cm. Find the perimeter of the trapezoid.

The segments DН and DE are tangent to the circle, then the segment OD is the diagonal of the СDA angle, and the OСD angle = 60/2 = 30.
By the property of a circle inscribed in a circle, the angle СOD = 90. Then in a right-angled triangle СOD, leg OС = 10 cm, lies opposite the angle 300, then the hypotenuse СD = 2 * OС = 2 * 10 = 20 cm.
In a right-angled triangle СOM, the angle СOM = 300, then the leg CM = OC / 2 = 10/2 = 5 cm.
The OM segment in the COM triangle will be equal to: OM = OС * Sin60 = 10 * √3 / 2 = 5 * √3 cm.
BC side = BM + CM = 5 * √3 + 5 cm.
Side AB = 2 * ОМ = 2 * 5 * √3 = 10 * √3 cm.
Side AD = AН + DН = AН + СD – CE = 5 * √3 + 20 – 5 = 15 + 5 * √3 cm.
The perimeter of the trapezoid is: AB + BC + СD + AD = 10 * √3 + 5 * √3 + 5 + 10 + 15 + 5 * √3 = 30 + 20 * √3 cm.
Answer: The perimeter of the trapezoid is 30 + 20 * √3 cm.



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