AD is a median of the `DeltaABC`.If AE and AFare medians of the triangles ABD respectively, and AD=`m_1,AE=m_2AF=m_3,` then find the value of `a^2/8`.

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)`