The perimeter of the rhombus is 20 cm, one diagonal with the side makes an angle of 75 degrees.

The perimeter of the rhombus is 20 cm, one diagonal with the side makes an angle of 75 degrees. Find the height of the rhombus.

All sides of the rhombus are equal, knowing the perimeter, we find the side:

a = P / 4 = 20/4 = 5 cm.

It is known that the diagonals of a rhombus are the bisectors of its corners, so if the diagonal makes an angle of 75 ° with the side of the rhombus, then one of the corners of the rhombus is 75 * 2 = 150 °.

The area of a rhombus is equal to the product of the square of its side by the sine of the angle between them:

S = a ^ 2 * sin α = 5 ^ 2 * sin 150 ° = 25 * 0.5 = 12.5 cm2.

On the other hand, the area of a rhombus can be found as the product of its side and height:

S = a * h.

Hence h = S / a = 12.5 / 5 = 2.5 cm – the height of the rhombus.