# In an isosceles triangle ABC, the medians intersect at point O. Find the distance from point O to the vertex B

In an isosceles triangle ABC, the medians intersect at point O. Find the distance from point O to the vertex B of this triangle if AB = AC = 13cm, BC = 10cm.

The median AD of an isosceles triangle ABC is also its height, then BD = CD = BC / 2 = 10/2 = 5 cm, and triangle ABD is rectangular, in which, according to the Pythagorean theorem, we determine the length of the leg AD.

AD ^ 2 = AB ^ 2 – BD ^ 2 = 169 – 25 = 144.

AD = 12 cm.

By the property of the medians of the triangle, the point of their intersection divides them in the ratio of 2/1, then AO = 2 * OD.

Then ОD = АD / 3 = 12/3 = 4 cm.

In a right-angled triangle BOD, according to the Pythagorean theorem, we determine the length of the hypotenuse OB.

ОВ ^ 2 = ОD ^ 2 + ВD ^ 2 = 16 + 25 = 41.

ОВ = √41 cm.

Answer: The distance from point O to peak B is √41 cm.

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