If $P( 2,p)$ is the mid-point of the line segment joining the points $A( 6,-5)$ and $B( -2,11)$. Find the value of $p$.

Given: A line segment joining the two points $A( 6,-5)$ and $B( -2,11)$ and a point $P( 2, p)$ as a mid-point of the line segment.

To do: To find the value of p.

If there is a line segment joining two points A$( x_{1} ,\ y_{1})$ and B$( x_{2} ,\ y_{2})$ then the mid-point $p( x,\ y)$

$( x,\ y) =(\frac{x_{1} +x_{2}}{2} ,\ \frac{y_{1} +y_{2}}{2} )$

$\therefore ( 2,\ p) =\left(\frac{6-2}{2} ,\ \frac{-5+11}{2}\right)$

$\Rightarrow ( 2,\ p) =\left(\frac{4}{2} ,\ \frac{6}{2}\right)$

$\Rightarrow ( 2,\ p) =( 2,3)$

$\therefore p=3$