Question

If \[P(1,2),Q(4,6),R(5,7)\] and then are vertices of parallelogram PQRS then
A.a=b , b=4
B.a=3, b=4
C.a=2, b=3
D.a=-3, b=5

Answer 1

Hint: Here vertices of parallelogram are given. But we need to take help of diagonals. Diagonals of parallelogram bisect each other at a point of its midpoint. So we can use the midpoint formula.
Formula used:
Midpoint formula: \[\dfrac{{{x_1} + {x_2}}}{2}\], \[\dfrac{{{y_1} + {y_2}}}{2}\]

Complete step-by-step answer:
Diagonals of parallelogram bisect each other at a point of its midpoint.
Formula used:
Midpoint formula: \[\dfrac{{{x_1} + {x_2}}}{2}\], \[\dfrac{{{y_1} + {y_2}}}{2}\]

Now since O is the midpoint of diagonals, we can use the midpoint formula of distance.
Let's take opposite vertices.
P and R as well as Q and S.
To find value of a:
\[\dfrac{{{x_1} + {x_2}}}{2}\]
\[ \Rightarrow \dfrac{{4 + a}}{2} = \dfrac{{1 + 5}}{2}\]
Cancelling 2 from both sides
\[
   \Rightarrow 4 + a = 1 + 5 \\
   \Rightarrow 4 + a = 6 \\
   \Rightarrow a = 6 - 4 \\
   \Rightarrow a = 2 \\
\]
To find value of b:
\[\dfrac{{{y_1} + {y_2}}}{2}\]
\[ \Rightarrow \dfrac{{6 + b}}{2} = \dfrac{{2 + 7}}{2}\]
Cancelling 2 from both sides
\[
   \Rightarrow 6 + b = 9 \\
   \Rightarrow b = 9 - 6 \\
   \Rightarrow b = 3 \\
\]
Hence option C is the correct option.
Additional information:
Parallelogram is having opposite sides parallel to each other.
Note: Don’t try using distance formula or any other formula here because the diagonals of parallelogram bisect each other will lead to solve the problem.