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  4. 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 a b.

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Q. 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 $a b$.

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Solution:

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