If in \( \triangle A B C, D \) is the mid-point of side \( B C \) and \( A B^{2}+A C^{2}=x\left(B D^{2}+A D^{2}\right) \) then the value \( x \) will be : \( \begin{array}{llll}\text { A) } 2 & \text { B) } 1 & \text { C) } 4 & \text { D) } 0\end{array} \) i) In the eriuation \( 3 x-9 v=17 \) if the value of \( x \) is 0 then the valuo of \( v \) ic.

In triangle \( ABC \) with \( D \) as the midpoint of side \( BC \), the relationship given is derived from the Apollonius's theorem. This theorem states that the sum of the squares of the sides of a triangle is related to the square of the third side and twice the square of half the remaining side. Thus, when we manipulate the equation, we find that \( x \) equals 2. As for solving \( 3x - 9v = 17 \) with \( x = 0 \), we substitute \( 0 \) into the equation. It simplifies to \( -9v = 17 \), thus \( v = -\frac{17}{9} \). So, remember, every equation sometimes leads to surprising outcomes!