## Precalculus (10th Edition)

Midpoint of both diagonals of a square with side length $s$ is same and it is at $\left( \frac{s}{2},\frac{s}{2} \right)$.
Consider the square with side length $s$ Let the vertices of square are $(0,0),\,(0,\,s),\,(s,\,0),\,(s,s)$ The mid-point formula: The coordinate of the mid-point of two points ${{P}_{1}}=({{x}_{1}},{{y}_{1}})$ and${{P}_{2}}=({{x}_{2}},{{y}_{2}})$is $M=(x,y)$ , where $x=\frac{{{x}_{1}}+{{x}_{2}}}{2},\,y=\frac{{{y}_{1}}+{{y}_{2}}}{2}$ Consider the diagonal with end points $(0,s)$and $(s,0)$ Therefore, the coordinate of mid-point of line joining two points ${{P}_{1}}=(0,s)$ and${{P}_{2}}=(s,0)$is $M=(x,y)$, where $x=\frac{0+s}{2},\,y=\frac{s+0}{2}$ Simplifying $x=\frac{s}{2},y=\frac{s}{2}$ The mid-point of the diagonal joining the points $(0,s)$and$(s,0)$is${{M}_{1}}=\left( \frac{s}{2},\frac{s}{2} \right)$ Consider the diagonal with end points $(0,0)$and $(s,s)$ Therefore, the coordinate of mid-point of the line joining two points ${{P}_{1}}=(0,0)$ and${{P}_{2}}=(s,s)$is $M=(x,y)$, where $x=\frac{0+s}{2},\,y=\frac{0+s}{2}$ Simplifying $x=\frac{s}{2},y=\frac{s}{2}$ The mid-point of the diagonal joining the points $(0,0)$and$(s,s)$ is${{M}_{2}}=\left( \frac{s}{2},\frac{s}{2} \right)$ Thus, the mid-point of the diagonals of square with length $s$ is same and it is $\left( \frac{s}{2},\frac{s}{2} \right)$ Therefore, the diagonals of square intersect at their mid-points