The perimeter of parallelogram ABCD is 30, and the angle BAD is 60.

| May 26, 2021 | education

The perimeter of parallelogram ABCD is 30, and the angle BAD is 60. A circle of radius √3 is inscribed in triangle BCD. Find the area of the parallelogram.

Let’s build the radii OK, ОМ and ОН. In a right-angled triangle CОК, the angle ОСК = 60/2 = 30, leg ОСК = R = √3 cm.Then tg30 = ОСК / СК.

CK = OK / tg30 = √3 / (1 / √3) = 3 cm.

By the property of tangents drawn from one point, CM = CK = 3 cm.

The parallelogram semi-perimeter is equal to: p = 30/2 = 15 cm, BC + СD = 15 cm.

Then BK + DМ = 15 – 3 – 3 = 9 cm, and therefore BH + DH = 9 cm, by the property of tangents.

The perimeter of the BCD triangle is: Pvsd = 9 + 15 = 24 cm, then the half-perimeter p = 24/2 = 12 cm.

Let’s define the area of ​​the triangle ВСD.

Svsd = p * r = 12 * √3 cm2.

Then Savsd = 2 * Svsd = 2 * 12 * √3 = 24 * √3 cm2.

Answer: The area of ​​the parallelogram is 24 * √3 cm2.



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