The angle at the base of an isosceles trapezoid is 30 degrees; the area of the trapezoid
| September 28, 2021 | education
The angle at the base of an isosceles trapezoid is 30 degrees; the area of the trapezoid is 72 cm2. Find the radius of the circle inscribed in the trapezoid.
By the property of the trapezoid into which the circle is inscribed, the sums of the lengths of the opposite sides are equal. AB + CD = BC + AD.
AD + BC = 2 * CD, since AB = CD.
The area of the trapezoid is:
S = (BC + AD) * CH / 2 = 2 * CD * CH / 2 = CD * CH. (1).
From triangle СНD, СН = СD * Sin30 = CD / 2.
Then S = CD * CD / 2 = C ^ 2/2.
CD ^ 2 = 72 * 2 = 144.
СD = 12 cm.
Substitute in equation 1. S = 12 * CH.
CH = 72/12 = 6 cm.
CH – the height of the trapezoid and the diameter of the circle, then R = 6/2 = 3 cm.
Answer: R = 3 cm.
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