Q. 144.2( 25 Votes )

# 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.

Answer :

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 AB^{2} + AC^{2} = 2AM^{2} + 2BM ^{2}

Putting values, we get

(13)^{2} + (13)^{2} = 2AM^{2} + 2(5)^{2}

⇒ 169 + 169 = 2AM^{2} + 50

⇒ 2AM^{2} = 288

⇒ AM^{2} = 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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