Diagonals AC and BD of square ABCD intersect at P. Coordinates of points B and D are $(9, -2)$ and $(1, 6)$ respectively. (i) Find the co-ordinates of point P. (ii) Find the length of the side of the square.
Generated by claude-sonnet-4-6 · 2026-06-15 10:30 · grounding rag
Model Answer
(i) In a square, diagonals bisect each other, so P is the mid-point of BD.
$$P = \left(\frac{9+1}{2},\ \frac{-2+6}{2}\right) = \left(5,\ 2\right)$$
(ii) Length of diagonal BD:
$$BD = \sqrt{(9-1)^2 + (-2-6)^2} = \sqrt{64+64} = 8\sqrt{2}$$
For a square, side $= \dfrac{\text{diagonal}}{\sqrt{2}} = \dfrac{8\sqrt{2}}{\sqrt{2}} = \mathbf{8}$ units.
Source: Chapter 7, Section 7.2 (Distance Formula) and Section 7.3 (Mid-point Formula)
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Explanation
- Part (i): Diagonals of a square bisect each other, so P is the midpoint of BD (and also of AC). Apply the midpoint formula directly.
- Part (ii): Use the distance formula to find diagonal BD, then use the relation: in a square, diagonal $= \text{side} \times \sqrt{2}$, so side $= \frac{BD}{\sqrt{2}}$.
- Examiners expect the midpoint formula shown clearly for (i) and the distance formula + the diagonal-side relation for (ii).