# GMAT Quant Questions | Geometry #1

#### Properties of Quadrilaterals | Geometry - Polygons | GMAT Sample Questions

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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### 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.

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.

#### 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 = $\sqrt{[5 - 4]^{2} + [4 - 5]^{2}}$ = $\sqrt{2}$ #### GMAT Online CourseTry it free! Register in 2 easy steps and Start learning in 5 minutes! #### Already have an Account? #### GMAT Live Online Classes Next Batch Oct 12, 2021 #### GMAT Coaching in Chennai Next Batch after Lockdown #### GMAT Coaching in Chennai Sign up for a Free Demo of our GMAT Classes in Chennai ##### Where is Wizako located? Wizako - GMAT, GRE, SAT Prep An Ascent Education Initiative 14B/1 Dr Thirumurthy Nagar 1st Street Nungambakkam Chennai 600 034. India ##### How to reach Wizako? Phone:$91) 44 4500 8484
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