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
2. Properties of quadrilaterals 2 concepts
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?
 Square
 Rectangle but not a square
 Rhombus
 Parallelogram but not a rhombus
Correct Answer Choice (3). The quadrilateral is a rhombus
Explanatory Answer

1. Properties of a square
 All 4 sides are equal. Opposite sides are parallel.
 All angles are equal.
 Opposite angles are equal and supplementary (sum of the angles equals 180^{o})
 Diagonals are equal and bisect each other at right angles
 A square is a cyclic quadrilateral
2. Properties of a Rectangle
 Opposite sides are parallel and equal.
 All angles are equal.
 Opposite angles are equal and supplementary
 Diagonals are equal and bisect each other but NOT at right angles
 A rectangle is a cyclic quadrilateral
3. Properties of a Parallelogram
 Opposite sides are parallel and equal.
 Opposite angles are equal
 Adjacent angles are supplementary
 Diagonals are NOT equal but bisect each other but NOT at right angles
 A parallelogram is NOT a cyclic quadrilateral
4. Properties of a Rhombus
 Opposite sides are parallel and equal.
 All 4 sides are equal.
 Opposite angles are equal
 Adjacent angles are supplementary
 Diagonals are NOT equal but bisect each other at right angles
 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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