Q1. [3]
ABCD is a rectangle formed by the points A $(-1, -1)$, B $(-1, 6)$, C $(3, 6)$ and D $(3, -1)$. P, Q, R and S are mid-points of sides AB, BC, CD and DA respectively. Show that diagonals of the quadrilateral PQRS bisect each other.
Previously asked in CBSE board exam
2024 30/1/1 Q26(B)
Generated by claude-sonnet-4-6 · 2026-06-15 10:32 · grounding rag
Model Answer
Finding mid-points using mid-point formula:
- P (mid-point of AB): $\left(\dfrac{-1-1}{2}, \dfrac{-1+6}{2}\right) = \left(-1, \dfrac{5}{2}\right)$
- Q (mid-point of BC): $\left(\dfrac{-1+3}{2}, \dfrac{6+6}{2}\right) = \left(1, 6\right)$
- R (mid-point of CD): $\left(\dfrac{3+3}{2}, \dfrac{6-1}{2}\right) = \left(3, \dfrac{5}{2}\right)$
- S (mid-point of DA): $\left(\dfrac{3-1}{2}, \dfrac{-1-1}{2}\right) = \left(1, -1\right)$
Mid-point of diagonal PR:
$$\left(\dfrac{-1+3}{2},\ \dfrac{\frac{5}{2}+\frac{5}{2}}{2}\right) = \left(1,\ \dfrac{5}{2}\right)$$
Mid-point of diagonal QS:
$$\left(\dfrac{1+1}{2},\ \dfrac{6+(-1)}{2}\right) = \left(1,\ \dfrac{5}{2}\right)$$
Since mid-points of both diagonals PR and QS are the same, i.e., $\left(1, \dfrac{5}{2}\right)$, the diagonals of quadrilateral PQRS bisect each other. $\hspace{2cm}\blacksquare$
Source: Chapter 7, Section 7.3 (Mid-point formula)
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Explanation
- The key idea: diagonals bisect each other ⟺ their mid-points coincide. This is the same logic used in Example 10 for a parallelogram.
- Find all four mid-points P, Q, R, S first, then find mid-points of diagonals PR and QS separately, and show they are equal.
- Show the working clearly — examiners award marks for each mid-point found and for the final comparison step.
- No need to prove it is a parallelogram; just showing the diagonals bisect is sufficient.
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