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Coordinate Geometry : Quadrilateral Properties

This aptitude practice question is a coordinate geometry question. This one is a medium level difficulty question and tests the following concepts: finding length of a line segment given coordinates of its end points; properties of quadrilaterals including square, rectangle, rhombus and parallelogram.

Question

Vertices of a quadrilateral ABCD are A(0, 0), B(4, 5), C(9, 9) and D(5, 4). What is the shape of the quadrilateral?

  1. Square
  2. Rectangle but not a square
  3. Rhombus
  4. Parallelogram but not a rhombus

  Correct Answer      Choice (3). The quadrilateral is a rhombus


Explanatory Answer

  • 1. Properties of a square

    1. All 4 sides are equal. Opposite sides are parallel.
    2. All angles are equal.
    3. Opposite angles are equal and supplementary (sum of the angles equals 180o)
    4. Diagonals are equal and bisect each other at right angles
    5. A square is a cyclic quadrilateral

    2. Properties of a Rectangle

    1. Opposite sides are parallel and equal.
    2. All angles are equal.
    3. Opposite angles are equal and supplementary
    4. Diagonals are equal and bisect each other but NOT at right angles
    5. A rectangle is a cyclic quadrilateral

    3. Properties of a Parallelogram

    1. Opposite sides are parallel and equal.
    2. Opposite angles are equal
    3. Adjacent angles are supplementary
    4. Diagonals are NOT equal but bisect each other but NOT at right angles
    5. A parallelogram is NOT a cyclic quadrilateral

    4. Properties of a Rhombus

    1. Opposite sides are parallel and equal.
    2. All 4 sides are equal.
    3. Opposite angles are equal
    4. Adjacent angles are supplementary
    5. Diagonals are NOT equal but bisect each other at right angles
    6. A rhombus is NOT a cyclic quadrilateral
  • Approach: Compute length of the sides

    1. Length of AB given A(0, 0) and B(4, 5) = \\sqrt{ (x_2 - x_1)^2 + (y_2 - y_1)^2 }\\)
    = \\sqrt{ (4 - 0)^2 + (5 - 0)^2 }\\) = \\sqrt41 \\)

    2. Length of BC given B(4, 5) and C(9, 9) = \\sqrt{ (9 - 4)^2 + (9 - 5)^2 }\\) = \\sqrt{41} \\)

    3. Length of CD given C(9, 9) and D(5, 4) = \\sqrt{ (5 - 9)^2 + (4 - 9)^2 }\\) = \\sqrt{41} \\)

    4. Length of DA given D(5, 4) and A(0, 0) = = \\sqrt{ (0 - 5)^2 + (0 - 4)^2 }\\) = \\sqrt{41} \\)

    The lengths of the four sides, AB, BC, CD and DA are all equal.
    ∴ the quadrilateral has to be either a square or a rhombus.

    Compute length of diagonals

    1. Length of diagonal AC given A(0, 0) and C(9, 9) = \\sqrt{ (9 - 0)^2 + (9 - 0)^2 }\\) = \\sqrt{162} \\)

    2. Length of diagonal BD given B(4, 5) and D(5, 4) = \\sqrt{ (5 - 4)^2 + (4 - 5)^2 }\\) = \\sqrt{2} \\)

    The diagonals are not equal and the sides are equal, the given quadrilateral is a Rhombus.

      The correct answer is Choice (3).

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