Quadrilateral

What is a Quadrilateral ？

Ø  It is a four-sided polygon with four angles Ø  The sum of interior angles is 360

Types of Quadrilateral

Square Rectangle Parallelogram Rhombus

Kite Trapezium Cyclic Quadrilateral

Rectangles and Squares

Rectangle

What is a rectangle? A quadrilateral where opposite sides are parallel.

Properties of a Rectangle

Ø  Opposite sides are congruent

Ø  Opposite sides are parallel

Ø  Internal angles are congruent

Ø  All internal angles are right angled (90 degrees)

Perimeter of a Rectangle

Step 1: Add up the sides

DONE

Example:

Perimeter: x + y + x + y

OR

2x + 2y

Area of a Rectangle

How?

Multiply the length with the width

Example:

Area = x . y

The Diagonal of a Rectangle

To find the length of diagonal on a rectangle:

Let diagonal = D

D squared = x squared + y squared

D

Properties of the Diagonals on Rectangles

Ø Diagonals do not intersect at right angles

Ø  Angles at the intersection can differ

Ø  Opposite angles at intersection are congruent

Square

What is a square? A quadrilateral with sides of equal length.

Properties of a Square

Ø The sides are congruent

Ø  Angles are congruent

Ø  Total internal angle is 360 degrees

Ø  All internal angles are right-angled (90 degrees)

Ø  Opposite angles are congruent

Ø  Opposite sides are congruent

Ø  Opposite sides are parallel

Perimeter of a Square

Step 1: Add up all the sides

DONE

Example:

Perimeter: a + a + a + a

OR 4a

Area of a Square How?

Multiply two of the sides together or just “SQUARE” the

length of one side

Example:

Area = a x a OR a squared

Diagonal of a Square

To find the length of the diagonal of the square, multiply the length of one side with the square root of 2.

Example: d = a

Properties of the Diagonals in Squares

Ø The diagonals intersect at

90 degree angles

(right-angled)

Ø  Diagonals are

perpendicular

Ø  Diagonals are congruent

Parallelogram

Opposite sides:

Ø Parallel

Ø Equal in length

Parallelogram

Opposite Interior Angles: Ø  Equal Ø  A = C

B = D

D + C = 180 Known as supplementary angles.

Parallelogram

The diagonals: Ø  Bisect each other Ø  Intersect each other at the half way point Each diagonal separates it into 2 congruent triangles.

Perimeter of Parallelogram

Perimeter = 2(a+b)

a  

b  

Area of Parallelogram

Area = Base x Height

Area of Parallelogram (Example)

Solution: 180o – 135o = 45o

sin 45o = h / 15 h = 10.6

18  

Area = Base x Height = 18 x 10.6 = 190.8

Rhombus

A flat shape with 4 equal straight sides that looks like a diamond.

Properties of Rhombus 1 - All sides are congruent (equal lengths). length AB = length BC = length CD = length DA = a. 2 - Opposite sides are parallel. AD is parallel to BC and AB is parallel to DC. 3 - The two diagonals are perpendicular. AC is perpendicular to BD. 4 - Opposite internal angles are congruent (equal sizes). internal angle A = internal angle C and internal angle B = internal angle D. 5 - Any two consecutive internal angles are supplementary : they add up to 180 degrees. angle A + angle B = 180 degrees angle B + angle C = 180 degrees angle C + angle D = 180 degrees angle D + angle A = 180 degrees

Area of Rhombus

Perimeter of Rhombus

Example :

Question : The lengths of the diagonals of a rhombus are 20 and 48 meters. Find the perimeter of the rhombus. Solution : • Below is shown a rhombus with the given diagonals. Consider the right triangle BOC and apply Pythagora's theorem as follows • BC 2 = 10 2 + 24 2 • and evaluate BC • BC = 26 meters. • We now evaluate the perimeter P as follows: • P = 4 * 26 = 104 meters.

CYCLIC QUADRILATERAL

A cyclic quadrilateral is a quadrilateral when there is a circle passing through all its four vertices.

Theorem 1: Sum of the opposite angles of a cyclic quadrilateral is 180°.

Example: ∠P + ∠R=180° and ∠S + ∠Q=180°

Theorem 2: Sum of all the angles of a cyclic quadrilateral is 360°.

Example: ∠P+∠Q+∠R+∠S = 360°

Proving Cyclic Quadrilateral Theorem

Area of Cyclic Quadrilateral

The area of the cyclic quadrilateral with sides a,b,c and d, and perimeter S= (a+b+c+d)/2 is given by Brahmagupta’s

Formula.

Kite

Ø  Two pairs of equal length - a & a, b & b, are adjacent to each other.

Ø  Diagonals are perpendicular to each other.

Perimeter = AB +BC + CD + DA

Area = ½ x d1 x d2

PERIMETER

AREA

Area = ½ x d1 x d2

= ½ x 4.8 x 10 = 24cm2

Area = ½ x d1 x d2 = ½ x (4+9) x

(3+3) = 39m2

EXAMPLE

Find the length of the diagonal of a kite whose area is 168 cm2 and one diagonal is 14 cm. Solution:

Given: Area of the kite (A) = 168 cm2 and one diagonal (d1) = 14 cm.

Area of Kite = ½ x d1 x d2

168 = ½ x 14 x d2

d2 = 168/7

d2 = 24cm

EXAMPLE

Trapezium

Properties: Only one pair of opposite side is parallel.

Area

Example

Class Activity

Question 1

•  Only one pair of opposite side is parallel.

Question 2 •  Opposite sides are parallel.

•  All sides are congruent (equal lengths).

•  Opposite internal angles are congruent (equal sizes).

•  The two diagonals are perpendicular.

•  Any two consecutive internal angles are supplementary : they add up to

180 degrees.

Question 3 Opposite sides: •  Parallel •  Equal in length

The diagonals: •  Bisect each other •  Intersect each other at the half way point Each diagonal separates it into 2 congruent triangles

Question 4 •  Two pairs of equal length - a & a, b & b, are adjacent

to each other. •  Diagonals are perpendicular to each other.

End of Presentation Thank You.