Question

Show that the points $(1,7),(4,2),(-1,-1)$ and $(-4,4)$ are the vertices of a square. $OR$ if $P(2,p)$ is the midpoint of the line segment joining the points $A(6,-5)$ and $B(-2,11)$, find the value of $p$?

Answer 1

Let the vertices are

$\begin{array}{c}M\left(1,7\right),N\left(4,2\right)\\ R\left(-1,-1\right),Q\left(-4,4\right)\\ MN=\sqrt{{\left(1-4\right)}^2+{\left(7-2\right)}^2}=\sqrt{9+25}=\sqrt{34}\\ NR=\sqrt{{\left(4+1\right)}^2+{\left(2+1\right)}^2}=\sqrt{25+9}=\sqrt{34}\\ RQ=\sqrt{{\left(-1+4\right)}^2+{\left(-1-4\right)}^2}=\sqrt{34}\\ MQ=\sqrt{{\left(1+4\right)}^2+{\left(7-4\right)}^2}=\sqrt{34}\end{array}$

all sides are equal, Hence MNRQ is a square

$P(2,P)$ is midpoint of

$\begin{array}{c}A\left(6,-5\right),B\left(-2,11\right)\\ P=\frac{-5+11}{2}\\ p=3\end{array}$