A trapezoid ABCD is inscribed in a circle of radius R with an acute angle at the base AD equal to a.

| February 10, 2021 | education

A trapezoid ABCD is inscribed in a circle of radius R with an acute angle at the base AD equal to a. It is known that the bisector of angle C passes through the center of the circumscribed circle. Find the area of the trapezoid

Since the trapezoid is inscribed in a circle, it is isosceles.

Since the OC is the bisector of the ВСD angle, the angle AСD = OСВ = ВСD / 2.

The OСD triangle is isosceles, since OС = OD = R.

Then the angle ODС = OСD, which means the angle ODС = OСD, and therefore the angle ВСD = 2 * ODС.

Let the angle ODС = X, then BCD= 2 * X.

X + 2 * X = 180.

X = ODС = 60.

ВСD angle = 120.

Then the triangle OСD is equilateral, OС = OD = СD = R = AD / 2 = a / 2.

The ВOС triangle is also equilateral, since OB = OB = R, and the angle ВOС = AOB = 600, then BC = OС / 2.

The area of ​​a trapezoid is equal to the sum of three equilateral triangles with side a.

Streug = a2 * √3 / 4 cm2.

Strap = 3 * Streug = 3 * a ^ 2 * √3 / 4 cm2.

Answer: The area of ​​the trapezoid is 3 * a2 * √3 / 4 cm2.



One of the components of a person's success in our time is receiving modern high-quality education, mastering the knowledge, skills and abilities necessary for life in society. A person today needs to study almost all his life, mastering everything new and new, acquiring the necessary professional qualities.