If the mid-point of the segment joining A (x, y + 1)
Question:

If the mid-point of the segment joining $A(x, y+1)$ and $B(x+1, y+2)$ is $C\left(\frac{3}{2}, \frac{5}{2}\right)$, find $x, y .$

Solution:

It is given that mid-point of line segment joining $\mathrm{A}(x, y+1)$ and $\mathrm{B}(x+1, y+2)$ is $\mathrm{C}\left(\frac{3}{2}, \frac{5}{2}\right)$

In general to find the mid-point $\mathrm{P}(x, y)$ of two points $\mathrm{A}\left(x_{1}, y_{1}\right)$ and $\mathrm{B}\left(x_{2}, y_{2}\right)$ we use section formula as,

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

So,

$\left(\frac{3}{2}, \frac{5}{2}\right)=\left(\frac{2 x+1}{2}, \frac{2 y+3}{2}\right)$

Now equate the components separately to get,

$\frac{2 x+1}{2}=\frac{3}{2}$

So,

$x=1$

Similarly,

$\frac{2 y+3}{2}=\frac{5}{2}$

So,

$y=1$