In rhombus ABCD, angle A is 60 degrees and segments BH and BF are heights. Calculate the area of the rhombus

| September 3, 2021 | education

In rhombus ABCD, angle A is 60 degrees and segments BH and BF are heights. Calculate the area of the rhombus if the perimeter of the HBF triangle is 12 centimeters.

The sum of the adjacent angles of the rhombus is 180, then the angle ADC = 180 – 60 = 120.

In the quadrangle ВНDF we define the angle НВF, taking into account that the angles ВНD = ВFD = 90, since ВН and BF are heights, then the angle HBF = 360 – 120 – 90 – 90 = 60. Since the rhombus has heights equal, then ВН = BF, and then the triangle BHF is equilateral. Then BH = BF = HF = P / 3 = 12/3 = 4 cm.

In a right-angled triangle ABН, we define the hypotenuse AB.

Sin60 = BH / AB.

AB = BH / Sin60 = 4 / (√3 / 2) = 8 / √3 cm.

Determine the area of the rhombus.

Savsd = BH * AD = 4 * 8 / √3 = 32 / √3 cm2.

Answer: The area of the rhombus is 32 / √3 cm2.



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