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The vertices of quadrilateral ABCD are A(1, 1), B(5, 3), C(4, 5) and D(0, 3). Using the distance formula, ABCD is:

Class 10Coordinate GeometryGeometry — Coordinates
  • Aa square
  • Ba rhombus that is not a square
  • Ca rectangle that is not a square
  • Da trapezium only

Step-by-Step Solution

AB = , BC = , CD = , DA = → opposite sides equal, so ABCD is a parallelogram. The diagonals AC = and BD = are EQUAL, which forces right angles → rectangle. It is not a square, because the adjacent sides and are different.
Answer: C) a rectangle that is not 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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