Question

If the points P(a, −11), Q(5, b), R(2, 15) and S(1, 1) are the vertices of a parallelogram PQRS, find the values of a and b.

Answer 1

The points are P(a, −11), Q(5, b), R(2, 15) and S(1, 1).

Join PR and QS, intersecting at O.
We know that the diagonals of a parallelogram bisect each other.
Therefore, O is the midpoint of PR as well as QS.
$$\begin{gathered} \mathrm{Midpoint}\mathrm{of}PR=\left(\frac{a+2}{2},\frac{-11+15}{2}\right)=\left(\frac{a+2}{2},\frac{4}{2}\right)=\left(\frac{a+2}{2},2\right) \\ \mathrm{Midpoint}\mathrm{of}QS=\left(\frac{5+1}{2},\frac{b+1}{2}\right)=\left(\frac{6}{2},\frac{b+1}{2}\right)=\left(3,\frac{b+1}{2}\right) \\ \mathrm{Therefore},\frac{a+2}{2}=3,\frac{b+1}{2}=2 \\ \Rightarrow a+2=6,b+1=4 \\ \Rightarrow a=6-2,b=4-1 \\ \Rightarrow a=4\text{and}b=3 \end{gathered}$$