# In an isosceles triangle, length of the congruent sides is 13 cm and its base is 10 cm. Find the distance between the vertex opposite the base and the centroid.

Let ABC be an isosceles triangle, In which AB = AC = 13 cm

And BC = 10 cm

Let AM be median on BC such that

Let P be centroid on median BC

To Find : AP [Distance between vertex opposite the base and centroid]

We know, By Apollonius theorem

In ΔABC, if M is the midpoint of side BC, then AB2 + AC2 = 2AM2 + 2BM 2

Putting values, we get

(13)2 + (13)2 = 2AM2 + 2(5)2

169 + 169 = 2AM2 + 50

2AM2 = 288

AM2 = 144

AM = 12 cm

Let P be the centroid

As, Centroid divides median in a ratio 2 : 1

AP : PM = 2 : 1

AP = 2PM

Now, AM = AP + PM

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