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?

- Square
- Rectangle but not a square
- Rhombus
- Parallelogram but not a rhombus
- Kite

From INR

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.

**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.

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})

Copyrights © 2016 - 23 All Rights Reserved by Wizako.com - An Ascent Education Initiative.

Privacy Policy | Terms & Conditions

GMAT^{®} is a registered trademark of the Graduate Management Admission Council (GMAC). This website is not endorsed or approved by GMAC.

GRE^{®} is a registered trademarks of Educational Testing Service (ETS). This website is not endorsed or approved by ETS.

SAT^{®} is a registered trademark of the College Board, which was not involved in the production of, and does not endorse this product.

Wizako - GMAT, GRE, SAT Prep

An Ascent Education Initiative

14B/1 Dr Thirumurthy Nagar 1st Street

Nungambakkam

Chennai 600 034. India

**Mobile:** (91) 95000 48484

**WhatsApp:** WhatsApp Now

**Email:** learn@wizako.com

Leave A Message