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.
In Euclidean plane geometry, a quadrilateral is basically a four sided polygon, with the internal angles adding up to 360°.
Types of quadrilaterals
Quadrilaterals are of five types based on their shapes. These are:
We’ll study each of these quadrilaterals in detail.
A parallelogram is a quadrilateral with opposite sides parallel (as seen in the figure below).
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.
If the length of a parallelogram is ‘l’, breadth is ‘b’ and height is ‘h’ then:
- Area = L * H
- Perimeter = 2(L+B)
A rectangle is a quadrilateral with all angles at right angles (360°/4 = 90°). (As seen in the figure below)
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°.
- If L = length and B = breadth, then
Length of the diagonal = √ (L2 + B2)
- Area = L * B
- Perimeter = 2(L+B)
A 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
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
A trapezium or a trapezoid is a quadrilateral that has only one pair of parallel sides called bases and two lateral sides called legs.
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:
|1||All sides are congruent||✕||✕||✓||✓|
|2||Opposite sides are parallel and congruent||✓||✓||✓||✓|
|3||All angles are congruent||✕||✓||✕||✓|
|4||Opposite angles are congruent||✓||✓||✓||✓|
|5||Diagonals are congruent||✕||✓||✕||✓|
|6||Diagonals are perpendicular||✕||✕||✓||✓|
|7||Diagonals bisect each other||✓||✓||✓||✓|
|8||Adjacent angles are supplementary||✓||✓||✓||✓|
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