In a right-angled triangle ABC C = 90, the medians CK and BM are mutually perpendicular

| June 2, 2021 | education

In a right-angled triangle ABC C = 90, the medians CK and BM are mutually perpendicular and intersect at point O, find the hypotenuse AB if OM √2.

Since CК and BM are the median of the triangle, at point O they are divided in the ratio of 2/1, then BО / MO = 2 / 1. BО = 2 * MO = 2 * √2 cm.

Then BM = MO + BО = √2 + 2 * √2 = 3 * √2 cm.

Since the medians intersect at right angles, the COM triangle is rectangular and is similar to the BCM triangle in an acute angle.

Then MB / CM = CM / OM.

CM ^ 2 = MB * OM = 3 * √2 * √2 = 6.

CM = √6 cm.

Then AC = 2 * CM = 2 * √6 cm.

In a right-angled triangle COM, according to the Pythagorean theorem, CO ^ 2 = CM ^ 2 – OM ^ 2 = 6 – 2 = 4.

CO = 2 cm.

From a right-angled triangle BOC, CB ^ 2 = CO ^ 2 + BO ^ 2 = 4 + 8 = 12.

СB = √12 = 2 * √3.

Then in a right-angled triangle ABC, AB ^ 2 = CB ^ 2 + AC ^ 2 = 12 + 24 = 36.

AB = 6 cm.

Answer: The length of the hypotenuse AB is 6 cm.



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