GMAT Quant Questions | Geometry #1

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 1: 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

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Video Explanation

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Explanatory Answer | GMAT Geometry

Approach to solve this GMAT Geometry Question: Compute the length of the sides and diagonals

The lengths of the four sides, AB, BC, CD, and DA are all equal to √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 √162 and the length of BD is √2.

Because the diagonals are not equal, whereas the sides are equal, the given quadrilateral is a Rhombus.

Choice C is the correct answer.

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Properties of Quadrilaterals

Properties of a square
A. All 4 sides are equal.
B. Opposite angles are equal and supplementary.
C. Diagonals are equal and bisect each other at right angles.
D. A square is a cyclic quadrilateral. It can be inscribed in a circle.

Properties of a Rhombus
A. All 4 sides are equal.
B. Opposite angles are equal but not supplementary.
C. Diagonals are not equal but bisect each other at right angles.
D. A rhombus is not a cyclic quadrilateral. A rhombus that can be inscribed in a circle is square.

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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})