The radius of the circle inscribed in the rhombus is 2.5 cm. One of the corners of the rhombus

| September 6, 2021 | education

The radius of the circle inscribed in the rhombus is 2.5 cm. One of the corners of the rhombus is 30 degrees. Find the diagonals of the rhombus.

The diameter НK of the inscribed circle in the rhombus coincides with the height of the rhombus. НK = 2 * R = 2 * 2.5 = 5 cm.

Let the side of the rhombus be X cm.

The area of the rhombus is equal to: Savsd = AB * HK = 5 * X cm2.

Also Savsd = AB * AD * Sin30 = X ^ 2 * (1/2) cm2.

Then 5 * X = X ^ 2/2.

X = 10 cm.

By the cosine theorem, in a triangle ABD, BD ^ 2 = AB ^ 2 + AD ^ 2 – 2 * AB * AD * Sin30 =

100 + 100 – 200 * √3 / 2 = 200 – 100 * √3 = 100 * (2 – √3).

ВD = 10 * √ (2 – √3) cm.

By the cosine theorem, in a triangle ABC, AC ^ 2 = AB ^ 2 + BC ^ 2 – 2 * AB * BC * Sin150 =

100 + 100 – 200 * (-√3 / 2) = 200 + 100 * √3 = 100 * (2 + √3).

AC = 10 * √ (2 + √3) cm.

Answer: The diagonals of the rhombus are 10 * √ (2 – √3) cm and 10 * √ (2 + √3) cm.



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