Question

Q.27. Show that the midpoint of the line segment joining the points P(0, -2)and R(0, 4) is also the midpoint of the line segment joining the points Q(3, 1) and S(- 3, 1). Also name the type of the quadrilateral PQRS.

Answer 1

$$\begin{gathered} \mathrm{We}\mathrm{know}\mathrm{that}\mathrm{mid}\mathrm{points}\mathrm{of}\mathrm{the}\mathrm{line}\mathrm{segment}\mathrm{joining}\mathrm{the}\mathrm{points}\left({\mathrm{x}}_1,{\mathrm{y}}_1\right)\mathrm{and}\left({\mathrm{x}}_2,{\mathrm{y}}_2\right)\mathrm{is}\mathrm{given}\mathrm{by} \\ \mathrm{mid}\mathrm{point}=\left(\frac{{\mathrm{x}}_1+{\mathrm{x}}_2}{2},\frac{{\mathrm{y}}_1+{\mathrm{y}}_2}{2}\right) \\ \mathrm{Now},\mathrm{coordinates}\mathrm{of}\mathrm{mid}\mathrm{point}\mathrm{of}\mathrm{line}\mathrm{segment}\mathrm{joining}\mathrm{points}\mathrm{P}\left(0,-2\right)\mathrm{and}\mathrm{R}\left(0,4\right)=\left(\frac{0+0}{2},\frac{-2+4}{2}\right)\equiv \left(0,1\right) \\ \mathrm{Now},\mathrm{coordinates}\mathrm{of}\mathrm{mid}\mathrm{point}\mathrm{of}\mathrm{line}\mathrm{segment}\mathrm{joining}\mathrm{points}\mathrm{Q}\left(3,1\right)\mathrm{and}\mathrm{S}\left(-3,1\right)=\left(\frac{3-3}{2},\frac{1+1}{2}\right)\equiv \left(0,1\right) \\ \mathrm{Since},\mathrm{mid}\mathrm{points}\mathrm{of}\mathrm{the}\mathrm{diagonals}\mathrm{PR}\mathrm{and}\mathrm{QS}\mathrm{of}\mathrm{quadrilateral}\mathrm{PQRS}\mathrm{coincide},\mathrm{then} \\ \mathrm{PQRS}\mathrm{is}\mathrm{a}\mathrm{parallelogram}. \end{gathered}$$