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The line segment joining A(1, −2) and B(7, 10) is cut into three equal parts by two points. Find the point of trisection that is closer to B.

Class 10Coordinate GeometryGeometry — Coordinates
  • A(3, 2)
  • B(4, 4)
  • C(5, 6)
  • D(4, 6)

Step-by-Step Solution

The trisection point nearer B divides AB in the ratio 2 : 1. ; , so the point is (5, 6). Using the ratio the other way round, 1 : 2, gives the OTHER trisection point (3, 2); (4, 4) is the midpoint, which is not a point of trisection at all; (4, 6) mixes the midpoint's with the correct .
Answer: C) (5, 6)

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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