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?

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

#### Explanatory Answer

Video explanation will be added soonThe 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

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

#### Properties of a Rhombus

- All 4 sides are equal.
- Opposite angles are equal but not supplementary.
- Diagonals are not equal but bisect each other at right angles.
- 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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