Q.423. Application of Apollonius theorem in geometry.

Question 423.
The lengths of sides a, b, c in a triangle are related as a² +b² =5c². Prove that medians drawn to sides "a" and "b" are mutually perpendicular.

Answer 423.
Let AM be the length of the median on side a
and BN be the length of the median on side b
Let G be the centroid where the medians AM and BN intersect.

By Apollonius theorem,
b^2 + c^2 = 2AM^2 + 2(a/2)^2
=> AM^2 = (1/2) (b^2 + c^2 - a^2/2)
=> AG^2 = (2/3)^2 * (1/2) (b^2 + c^2 - a^2/2)
=> AG^2 = (2/9) (b^2 + c^2 - a^2/2)

Similarly, BG^2 = (2/9) (c^2 + a^2 - b^2/2)

=> AG^2 + BG^2
= (2/9) (b^2 + c^2 - a^2/2 + c^2 + a^2 - b^2/2)
= (2/9) (a^2/2 + b^2/2 + 2c^2)
= (2/9) [(1/2) (5c^2) + 2c^2] ... (plugging a^2 + b^2 = 5c^2 given)
= c^2.

=> triangle AGB is a right triangle and AG is perpendicular to BG
=> medians drawn to sides a and b are mutually perpendicular.

Link to YA!