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.

Question

Philoid · Accepted 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 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