# GMAT Practice Question - Quadrilaterals

Concept: Quadrilateral shapes and properties. Basics of coordinate geometry.

This GMAT quant practice question is a problem solving question in Geometry. Concept: Properties of polygons (quadrilaterals in specific) and their shapes in geometry and elementary concepts in coordinate geometry - finding length of a line segment, if the coordinates of its end points are known.

#### 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
5. Kite

Video explanation will be added soon

The lengths of the four sides, AB, BC, CD and DA are all equal to $$sqrt{41}\\$.$Computation given in the last paragraph)
Hence, the given quadrilateral is either a Rhombus or a Square.

How to determine whether the quadrilateral is a square or a rhombus?
The diagonals of a square are equal. The diagonals of a rhombus are unequal.
Compute the lengths of the two diagonals AC and BD.

The length of AC is $$sqrt{162}\\$ and the length of BD is $\sqrt{2}\\$. As the diagonals are not equal and the sides are equal, the given quadrilateral is a Rhombus. Choice C is the correct answer. #### Properties of a square 1. All 4 sides are equal. 2. Opposite angles are equal and supplementary. 3. Diagonals are equal and bisect each other at right angles. 4. A square is a cyclic quadrilateral. It can be inscribed in a circle. #### Properties of a Rhombus 1. All 4 sides are equal. 2. Opposite angles are equal but not supplementary. 3. Diagonals are not equal but bisect each other at right angles. 4. A rhombus is not a cyclic quadrilateral. A rhombus that can be inscribed in a circle is square. #### Computation of length of sides and diagonals of the polygon Vertices of the quadrilateral are A$0, 0), B(4, 5), C(9, 9) and D(5, 4)
Side AB = $\sqrt{[4 - 0]^2 + [5 - 0]^2} = \sqrt{41} \\$.
Side BC = $\sqrt {[9 - 4]^2 + [9 - 5]^2} = \sqrt{41} \\$.
Side CD = $\sqrt{[5 - 9]^2 + [4 - 9]^2} = \sqrt{41} \\$.
Side DA = $\sqrt{[0 - 5]^2 + [0 - 4]^2} = \sqrt{41}\\$.

Diagonal AC = $\sqrt{[9 - 0]^2 + [9 - 0]^2} = \sqrt{162}\\$.
Diagonal BD = $\sqrt{[5 - 4]^2 + [4 - 5]^2} = \sqrt{2} \\$.

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