Question

If the points p (0,k),Q(8,3),R(6,7)and S(-2,3)are the vertices of a rectangle find the value of k

Answer 1

$\underline{\textsf{Given:}}$

$\textsf{The points P(0,k),Q(8,3),R(6,7) and S(-2,3) are vertices of a rectangle}$

$\underline{\textsf{To find:}}$

$\textsf{The value of k}$

$\underline{\textsf{Solution:}}$

$\textsf{We know that,}$

$\textsf{"Diagonals of rectangle bisect each other"}$

$\implies\textsf{Midpoint of diagonal PR=Midpoint of diagonal QS}$

$\implies\displaystyle\mathsf{(\dfrac{0+6}{2},\dfrac{k+7}{2})=(\dfrac{8+(-2)}{2},\dfrac{3+3}{2})}$

$\implies\displaystyle\mathsf{(\dfrac{6}{2},\dfrac{k+7}{2})=(\dfrac{6}{2},\dfrac{6}{2})}$

$\implies\displaystyle\mathsf{(3,\dfrac{k+7}{2})=(3,3)}$

$\textsf{Equating corresponding coordinates on bothsides, we get}$

$\mathsf{\dfrac{k+7}{2}=3}$

$\mathsf{k+7=6}$

$\mathsf{k=6-7}$

$\implies\boxed{\mathsf{k=-1}}$

$\underline{\textsf{Answer:}}$

$\textsf{The value of k is -1}$

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