In a right-angled triangle ABC with right angle C, the medians CC1 and BB1 are perpendicular to each other.

| August 16, 2021 | education

In a right-angled triangle ABC with right angle C, the medians CC1 and BB1 are perpendicular to each other. Find the length of the larger of these medians if the length of the third median is AA1 = 3√3.

Since the medians CH and BK intersect at right angles, the COB triangle is rectangular, and OM is its median drawn to the hypotenuse.

Then CM = BM = OM.

According to the property of the medians, the point of their intersection divides them in a ratio of 2/1, then CM = BM = OM = AM / 3 = √3 cm. BC = 2 * OM = 2 * √3 cm.

In a right-angled triangle AFM, according to the Pythagorean theorem, AC ^ 2 = AM ^ 2 – CM ^ 2 = 27 – 3 = 24.

AC = √24 = 2 * √6 cm.

In a right-angled triangle ABC, AB ^ 2 = AC ^ 2 + BC ^ 2 = 24 + 12 = 36.

AB = 6 cm.

Then CH = AB / 2 = 3 cm.

In a right-angled triangle СBК СК = АС / 2 = √6 cm.

Then BK ^ 2 = CK ^ 2 + BC ^ 2 = 6 + 12 = 18.

BK = 3 * √2 cm.

Answer: The length of the greater median is equal to VK = 3 * √2 cm.



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