About Different Types of Quadrilateral
What is a Quadrilateral?
As we already know, a quadrilateral is a polygon that has four sides.
Depending upon the length of sides and the measure of the angles, quadrilaterals can be of different types.
Types of Quadrilaterals
1. Square:
A square is a quadrilateral with all four sides congruent and each angle is a right angle, that is, 90°. The diagonals are of equal length and bisects each other at 90°. The opposite sides are parallel to each other.
In the above figure, ABCD is a square
AB = BC = CD = AD
∠A = ∠B = ∠C = ∠D = 90°
AC = BD
AC \(\perp\) BD
2. Rectangle:
A rectangle is a quadrilateral with each angle being a right angle, that is, 90°. The opposite sides are congruent and are parallel to each other. The diagonals of a rectangle are equal in length.
In the above figure,
PQ = SR and PS = QR
PQ \(\parallel\) SR and PS \(\parallel\) QR
∠P = ∠Q = ∠R = ∠S = 90°
3. Rhombus:
A rhombus is a quadrilateral in which all the four sides are congruent. The opposite sides are parallel to each other and the diagonals bisect each other at right angles. Additionally, the opposite angles are equal in measure.
In the above figure,
EF = FG = GH = HE
∠E = ∠G and ∠H = ∠F
4. Parallelogram:
A parallelogram is a quadrilateral with both pairs of opposite sides being congruent and parallel to each other. Opposite angles are equal to each other.
In the above figure,
AB = CD and BC = AD
AB \(\parallel\) CD and BC \(\parallel\) AD
∠A = ∠C and ∠B =∠D
5. Trapezoid:
A trapezoid is a quadrilateral with exactly a pair of opposite sides that are parallel to each other. The parallel sides are called bases, while the non-parallel sides are called legs.
In the above figure,
PQ \(\parallel\) SR, since one side is parallel to the opposite side.
PQ and SR are called the bases, while PS and QR are called the legs.
6. Kite:
A kite is a quadrilateral with two pairs of adjacent sides that are congruent.
The diagonals of a rhombus bisect each other at 90°.
In the above figure,
PM = NM and OP = ON
MO \(\perp\) PN
Rapid Recall
Solved Examples
Example 1: If the perimeter of a rectangle is 88 cm and width is 4 cm, find its area.
Solution:
Given: Perimeter of rectangle = 88 cm
Width = 4 cm
Perimeter of rectangle P = 2 x (Length + Width)
88 = 2 × (Length + 4) [Substitute the values of perimeter and width]
44 = length + 4 [Divide each side by 2]
40 = length [Subtract 4 both sides]
Area of rectangle A = Length x Width
A = 40 × 4 [Substitute the values of length and width]
A = 160
Therefore, the area of the rectangle is 160 square centimeters.
Example 2: The area of a rhombus is 340 square units, and one of the diagonals is 17 units. Find the length of another diagonal.
Solution:
Let the length of other diagonals be d
Area of rhombus \(A=\frac{1}{2}~\times\) (Product of length of diagonals)
\(340=\frac{1}{2}~\times~ (17~\times~d)\) [Substitute the values of area and length of one diagonal]
340 × 2 = 17 × d [Multiply each side by 2]
\(\frac{680}{17}=d\) [Divide each side by 17]
40 = d
Therefore, the length of another diagonal is 40 units.
Example 3: John has to make a kite with diagonals of length 30 units and 50 units. How much will he spend on paper if the cost of paper is 0.004 dollars per square unit?
Solution:
To find the cost of paper, first we will find the area of the kite and then multiply it to the price per square units.
Area of kite = \(\frac{1}{2}~\times\) (Product of length of diagonals)
= \(\frac{1}{2}\) × (30 × 50) [Substitute the value of length of diagonals]
= \(\frac{1}{2}\) × (1500) [Simplify]
= 750 square units
So, John will require 750 square units of paper to make a kite.
Money spent on paper = Area of kite x Dollars per unit area
= 750 × 0.004 [Substitute the values]
= 750 × \(\frac{4}{1000}\) [Simplify]
= 3 Dollars
Therefore, John will spend $3 on paper to make the kite.
There are 6 types of quadrilateral:
- Square
- Rectangle
- Rhombus
- Parallelogram
- Trapezium
- Kite
Both a square and a rhombus have all the four sides equal. However, in a square each angle is a right angle, and in a rhombus the opposite angles are equal.
Yes, every rectangle is a parallelogram as it has opposite sides parallel and equal. But every parallelogram is not a rectangle, as each angle in a parallelogram is not a right angle.
Therefore, every rectangle is a parallelogram but every parallelogram is not a rectangle.
A kite is a quadrilateral that has two adjacent sides that are equal.
An angle sum property of a quadrilateral states that the sum of all the interior angles is 360 degrees.