Question

A rectangle has two opposite vertices $(1,2)$ and $(5,5)$. If other vertices are $(3,a)$ and $(3,b)$, then find the value of $ab$.

Answer 1

Answer: 6

Let $A=(1,2),C=(5,5)$ and $B,D$ be other two opposite vertices.
Let $M$ be the point of intersection of diagonals $AC$ and BD. Then
$M=\left(3,\frac{7}{2}\right)$
$|AC|=\sqrt{(5-1{)}^2+(5-2{)}^2}=5=|BD|$

$\Rightarrow |BM|=|MD|=\frac{5}{2}$
$\Rightarrow B=\left(3,\frac{7}{2}-\frac{5}{2}\right)=(3,1)$
and $D=\left(3,\frac{7}{2}+\frac{5}{2}\right)=(3,6)$
$\Rightarrow a=1,b=6,$
$\Rightarrow ab=6$