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The segment joining A(2, −3) and B(10, 9) is divided into FOUR equal parts by the points P, Q and R, named in order starting from A. Find the coordinates of Q.

Class 10Coordinate GeometryGeometry — Coordinates
  • A(6, 3)
  • B(4, 0)
  • C(8, 6)
  • D(6, 6)

Step-by-Step Solution

and divide AB in the ratios 1 : 3, 2 : 2 and 3 : 1. The middle point therefore splits AB in the ratio 1 : 1 — it is simply the midpoint: . For completeness and = (8, 6), so choosing (4, 0) answers for the point nearest A rather than the middle one.
Answer: A) (6, 3)

Key idea: Recognising shapes from coordinates

**Common tasks**:

- **Equilateral triangle**: all three sides equal length (use distance formula).
- **Isosceles**: two sides equal.
- **Right triangle**: side lengths satisfy Pythagoras .
- **Square**: all four sides equal AND adjacent sides perpendicular (or use diagonals: equal and bisect at right angles).
- **Rhombus**: all four sides equal but not necessarily right-angled.
- **Parallelogram**: opposite sides equal (use distance formula on all 4 sides — opposite pairs equal).
- **Rectangle**: opposite sides equal AND diagonals equal.

The procedure is always: compute side lengths via distance formula → match against the criteria for each shape.

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