# From the point of intersection of the diagonals of the rhombus, a perpendicular is drawn

**From the point of intersection of the diagonals of the rhombus, a perpendicular is drawn which divides the side of the rhombus into segments 18 cm and 32 cm. Find the length of the diagonals of the rhombus.**

The diagonals of the rhombus at the point of their intersection are halved and intersect at right angles. Then AO = OC = AC / 2, OВ = OD = ВD / 2, and the triangle AOD is rectangular.

The height of OH is drawn from the top of the right angle to the hypotenuse, then OH ^ 2 = AH * DH = 32 * 18 = 576.

OH = 24 cm.

In a right-angled triangle AOН, according to the Pythagorean theorem, AO ^ 2 = AH ^ 2 + OH ^ 2 = 1024 + 576 = 1600. AO = 40 cm, then AC = 2 * 40 = 80 cm.

In a right-angled triangle DOH, according to the Pythagorean theorem, DO ^ 2 = DН ^ 2 + OH ^ 2 = 324 + 576 = 900. DO = 30 cm, then BD = 2 * 30 = 60 cm.

Answer: The diagonals of the rhombus are 60 cm and 80 cm.