Let `A D` be a median of the ` A B Cdot` If `A Ea n dA F` are medians of the triangle `A B Da n dA D C` , respectively, and `A D=m_1,A E=m_2,A F=m_3,` then`(a^2)/8` is equal to `m2 2+m3 2-2m1 2` `m1 2+m2 2-2m3 2` `m1 2+m3 2-2m2 2` `non eoft h e s e`

Options

A. `m_(2)^(2) + m_(3)^(2) - 2m_(1)^(2)`
B. `m_(1)^(2) + m_(2)^(2) - 2m_(3)^(2)`
C. `m_(1)^(2) + m_(3)^(2) - 2m_(2)^(2)`
D. none of these

Answer

A

Explanation

In `DeltaABC, AD^(2) = m_(1)^(2) = (c^(2) + b^(2))/(2) -(a^(2))/(4)`
In `DeltaABD, AE^(2) = m_(2)^(2) = (AD^(2) + c^(2))/(2) - (((a)/(2))^(2))/(4)` [Apolloniun theorem]
In `DeltaADC, AF^(2) = m_(3)^(2) = (AD^(2) + b^(2))/(2) -(((a)/(2))^(2))/(4)`
`:. m_(2)^(2) + m_(3)^(2) = AD^(2) + (b^(2) + c^(2))/(2) - (a^(2))/(8)`
`= m_(1)^(2) + m_(1)^(2) + (a^(2))/(4) -(a^(2))/(8) = 2m_(1)^(2) + (a^(2))/(8)`
or `m_(2)^(2) + m_(3)^(2) - 2m_(1)^(2) = (a^(2))/(8)`