The diagonals of the rhombus ABCD intersect at point O. On the side AB, point K is taken so that OK is perpendicular

| February 19, 2021 | education

The diagonals of the rhombus AВСD intersect at point O. On the side AB, point K is taken so that OK is perpendicular to AB, AK = 2cm, ВK = 8cm. Find the diagonals of the rhombus.

The diagonals of the rhombus, at the intersection point, are divided in half and intersect at right angles, AC = 2 * AO, ВD = 2 * OВ, and triangle AOB is rectangular.

Since OK is perpendicular to AB, then OK is the height drawn from a right angle to the hypotenuse.

Then OK ^ 2 = AK * ВK = 2 * 8 = 16.

OK = 4 cm.

In a right-angled triangle AOK, according to the Pythagorean theorem, OA ^ 2 = AK ^ 2 + OK ^ 2 = 4 + 16 = 20. OA = √20 = 2 * √5 cm. AC = 4 * √5 cm.

In a right-angled triangle ВOK, according to the Pythagorean theorem, OB ^ 2 = ВK ^ 2 + OK ^ 2 = 64 + 16 = 80. OA = √80 = 4 * √5 cm. AC = 8 * √5 cm.

Answer: The diagonals of the rhombus are 4 * √5 cm and 8 * √5 cm.



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