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.
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