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Four survey pegs are driven in at A(0, 1), B(4, 4), C(1, 8) and D(−3, 5). Taken in order, what shape do they enclose?

Class 10Coordinate GeometryGeometry — Coordinates
  • AA rhombus whose diagonals are unequal
  • BA square
  • CA rectangle whose adjacent sides are unequal
  • DJust a parallelogram, nothing more

Step-by-Step Solution

AB = , BC = , CD = , DA = — all four sides are 5. Diagonals: AC = and BD = , equal as well. Equal sides plus equal diagonals → square. Because this square is tilted, sketching it and guessing 'rhombus with unequal diagonals' is the common error; only the diagonal check settles it.
Answer: B) A square

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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