NIMCET 2021 — Mathematics PYQ

To solve for the missing vertex $S(a,b)$, we use the fundamental property of parallelograms: The diagonals bisect each other.

1. Identify the Diagonals

In a parallelogram $PQRS$, the diagonals are $PR$ and $QS$. Since they bisect each other, they share the same midpoint.

2. Apply the Midpoint Formula

The midpoint of a line segment joining $(x_1,y_1)$ and $(x_2,y_2)$ is given by:

• Midpoint of diagonal $PR$:

  Using $P(1,2)$ and $R(5,7)$:

  $${\text{Midpoint}}_{PR}=\left(\frac{1+5}{2},\frac{2+7}{2}\right)=\left(\frac{6}{2},\frac{9}{2}\right)=(3,4.5)$$

• Midpoint of diagonal $QS$:

  Using $Q(4,6)$ and $S(a,b)$:

  $${\text{Midpoint}}_{QS}=\left(\frac{4+a}{2},\frac{6+b}{2}\right)$$

3. Solve for $a$ and $b$

Since the midpoints are identical, we equate the coordinates:

For the x-coordinate:

For the y-coordinate:

Final Answer:

The coordinates of vertex $S$ are $(2,3)$, so $a=2$ and $b=3$. The correct option is A.