Quadrilateral

What is a Quadrilateral ？ It is a four-sided polygon with four angles The sum of interior angles is 360

Types of Quadrilateral

Square Rectangle Parallelogram Rhombus

Kite TrapeziumCyclic Quadrilateral

Rectangles and Squares

RectangleWhat is a rectangle?

A quadrilateral where opposite sides are parallel.

Properties of a Rectangle Opposite sides are congruent Opposite sides are parallel Internal angles are congruent All internal angles are right angled (90 degrees)

Perimeter of a RectangleStep 1: Add up

the sides

DONEExample:

Perimeter: x + y + x + y

OR

2x + 2y

Area of a RectangleHow?

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

Diagonals do not intersect at right angles

Angles at the intersection can differ

Opposite angles at intersection are congruent

SquareWhat is a square?

A quadrilateral with sides of equal length.

Properties of a Square

The sides are congruent Angles are congruent Total internal angle is 360 degrees All internal angles are right-angled (90 degrees) Opposite angles are congruent Opposite sides are congruent Opposite sides are parallel

Perimeter of a SquareStep 1:

Add up all the sides

DONEExample:

Perimeter:a + a + a + a

OR

4a

Area of a SquareHow?

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

The diagonals intersect at 90 degree angles (right-angled) Diagonals are perpendicular Diagonals are congruent

Parallelogram

Opposite sides: Parallel Equal in length

Parallelogram

Opposite Interior Angles: Equal

A = C

B = D

D + C = 180

Known as supplementary angles.

Parallelogram

The diagonals: Bisect each other 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

RhombusA flat shape with 4 equal straight sides

that looks like a diamond.

Properties of Rhombus1 - 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

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

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 3Opposite 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 PresentationThank You.