Question

Let $P(1,2)$, $Q(a,b)$, $R(5,7)$ and $S(2,3)$ be the vertices of a parallelogram $PQRS$. Then ________.

• A. $a=4, b=2$
• B. $a=6, b=2$
• C. $a=6, b=4$
• D. $a=3, b=2$
• E. $a=4, b=6$

Hint:

For a parallelogram $PQRS$, $P + R = Q + S$ (using coordinate addition).

Answer 1

Answer: (E)

Step 1: Concept
In a parallelogram, the diagonals bisect each other. Thus, the midpoint of diagonal $PR$ is the same as the midpoint of diagonal $QS$.

Step 2: Meaning
Midpoint of $PR = \left(\frac{1+5}{2}, \frac{2+7}{2}\right) = (3, 4.5)$.

Step 3: Analysis
Midpoint of $QS = \left(\frac{a+2}{2}, \frac{b+3}{2}\right)$.
Equating: $\frac{a+2}{2} = 3 \implies a+2=6 \implies a=4$.
Equating: $\frac{b+3}{2} = 4.5 \implies b+3=9 \implies b=6$.

Step 4: Conclusion
The values are $a=4$ and $b=6$. Final Answer: (E)