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GMAT / GRE Tutorial Series: Quadrilaterals

Quadrilaterals: GMAT / GRE Mathematics Tutorial

This tutorial is based on geometry concepts. We will study the types and properties of five quadrilaterals: Parallelogram, rectangle, square, rhombus and trapezium.

Basic definition

In Euclidean plane geometry, a quadrilateral is basically a four sided polygon, with the internal angles adding up to 360°.

Quadrilateral

Quadrilateral

Types of quadrilaterals

Quadrilaterals are of five types based on their shapes. These are:

  • Parallelogram
  • Rectangle
  • Square
  • Rhombus
  • Trapezium

quadrilateral typesWe’ll study each of these quadrilaterals in detail.

Parallelogram

A parallelogram is a quadrilateral with opposite sides parallel (as seen in the figure below).

Parallelogram

Properties of a parallelogram

  • Opposite angles are equal.
  • Opposite sides/facing sides are parallel and of equal length.
  • Adjacent angles are supplementary, i.e. they add up to 180°.
  • Each diagonal divides the parallelogram into two equal triangles. The two diagonals bisect each other.
  • If one of the angles of a parallelogram is a right angle then the parallelogram is a rectangle.

Important formulas

If the length of a parallelogram is ‘l’, breadth is ‘b’ and height is ‘h’ then:

  • Area = L * H
  • Perimeter = 2(L+B)

Parallelogram


Rectangles

A rectangle is a quadrilateral with all angles at right angles (360°/4 = 90°). (As seen in the figure below)

Rectangle

 

Properties of a Rectangle

  • Opposite sides are both parallel and equal to each other.
  • All angles are 90°.
  • The diagonals are equal and bisect each other (divide each other equally).
  • Opposite angles formed at the point where diagonals meet are equal.
  • A rectangle is a parallelogram with all the angles 90°.

Important formulas

  • If L = length and B = breadth, then

Length of the diagonal = √ (L2 + B2)

  • Area = L * B
  • Perimeter = 2(L+B)

Rectangle


Squares

SquareA square is a quadrilateral in which all the sides are equal (as seen in the figure above).

Properties of a square

  • All sides and angles are equal.
  • Opposite sides are parallel to each other.
  • The diagonals are equal and bisect each other at right angles.
  • A square is a type of parallelogram in which all the angles and sides are equal.
  • Also, a parallelogram becomes a square when the diagonals are equal and right bisectors of each other.

Important formulas for Squares

  • If ‘L’ is the length of the side of a square then length of the diagonal = L √2.
  • Area = L2.
  • Perimeter = 4L

Square


Rhombus

Rhombus

A rhombus is a quadrilateral having equal sides and opposite sides are parallel to each other. (As seen in the figure above)

Properties of a Rhombus

  • All sides are equal
  • Opposite angles are equal
  • The diagonals bisect each other at right angles
  • Adjacent angles are supplementary (e.g. ∠A + ∠B = 180°)
  • A rhombus is also a parallelogram that has diagonals that are perpendicular to each other
  • A rhombus with right angles is a square

Important formulas for a Rhombus

If a and b are the lengths of the diagonals of a rhombus,

  • Area = (a* b) / 2
  • Perimeter = 4L

Rhombus


Trapezium

A trapezium or a trapezoid is a quadrilateral that has only one pair of parallel sides called bases and two lateral sides called legs.

Trapezium

Properties of a Trapezium

  • Only the bases of the trapezium are parallel to each other (MN ⫽ OP).
  • None of the sides, angles and diagonals are equal.

Important Formulas for a Trapezium

  • Area = (1/2) h (L+L2)
  • Perimeter = L + L1 + L2 + L3 

Summary of properties

Summarizing what we have learned so far for quick reference and recollection:

S.No.PropertyParallelogramRectangleRhombusSquare
1All sides are congruent
2Opposite sides are parallel and congruent
3All angles are congruent
4Opposite angles are congruent
5Diagonals are congruent
6Diagonals are perpendicular
7Diagonals bisect each other
8Adjacent angles are supplementary

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