A circle of radius 5 is circumscribed about trapezoid ABCD with bases AD and BC. The center of the circumscribed

| September 3, 2021 | education

A circle of radius 5 is circumscribed about trapezoid ABCD with bases AD and BC. The center of the circumscribed circle lies on the base AD. Base BC is 6. Find the diagonal AC of the trapezoid.

Since a circle is described around the trapezoid, this trapezoid is isosceles.

Let’s build the height of the CH. Since the trapezium is isosceles, the height CH divides the base of AD into two segments, the length of the smaller of which is equal to the half-difference of the lengths of the bases of the trapezoid. DH = (AD – BC) / 2.

Since the center of the circle lies on the basis of AD, then AD = 2 * R = 2 * 5 = 10 cm.

DH = (10 – 6) / 2 = 4/2 = 2 cm, then AH = AD – DH = 10 – 2 = 8 cm.

Triangle ACD is rectangular, since the inscribed angle ACD is based on the diameter of the circle. The height of CH is drawn from the top of the right angle to the hypotenuse, then CH ^ 2 = AH * DH = 8 * 2 = 16 cm.

CH = 4 cm.

Determine the area of ​​the trapezoid.

Savsd = (ВС + АD) * СН / 2 = (6 + 10) * 4/2 = 32 cm2.

Answer: The area of ​​the trapezoid is 32 cm2.



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