Area of Geometric Figure

8/14/2019 Area of Geometric Figure

1/19

Geometrical representation of some formulae

Presented by :-

Ranjit Singh M.Sc(maths)Govt.Girls Secondary school, Amloh (Fatehgarh Sahib)


2/19

Objectives

1) Mathematics is offenly considered as

the tough and rough subject, first of all we

have to create the interest of the students

in mathematics.

2) To enable the students to give new

concepts in the field of mathematics


3/19

Previous knowledge testing 1) what is square?

2) what is rectangle?

3) How can you find the area of any geometrical figure?

4) Can we use some geometrical figures to obtain the basic formulaelike

(x+a)(x+b) = x2+ (a+b)x + ab

(a+b)2= a2+2ab+b2

( a-b)2 = a2 2ab + b2

Lets try


4/19

Announcement of the topic

Derivation of the formulae

1) (x+ a) (x+b)= x2

+(a+b)x+ab

2) (a+b)2= a2+b2+2ab

3) (a-b)2=a2+b2-2ab


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DERIVATION OF THE FORMULA

(x+a) (x+b)=x2+(a+b)x+ab Take a rectangular cardboard ABCD Take (x+a) and (x+b) its lengths Now take AG=x and GD=b as shown in the figure Draw EFAD and GH AB Area of cardboard=l*b=(x+a) (x+b)

ab

ax

C

BA

x2

E

bx

D F

G H

a

x

bb

x

a

x

x

O

(x+a)

(x+b)


6/19

Derivation of formula

(x +a) (x + b) = x2 + (a+b) x + ab

Take a rectangular cardboard ABCD and take (x+a) and (x+b) itssides as shown .

C

BA

D

H

(x+a)

(x+b


7/19

Derivation of formula

(x +a) (x + b) = x2 + (a+b) x + ab

Divide rectangular cardboard ABCD in four portions . The area ofeach portion will be as shown in figure

ab

ax

C

B

O

A

x2

E

bx

D F

G H

a

x

bb

x

a

x

x

(x+a)

(x+b


8/19

Combine the area of each portion to get the desired result

x2+ ax + bx + ab = x2+ (a+b) x+ ab

(x+a) (x+b) = x2

+ (a+b)x + ab

H

C

OG

EA x

x

F

G O

D

x

b

O

B

H

E a

x

F

O a

b abbx

x2 ax


9/19

Derivation of the formula

(a+b)2 = a2+2ab+b2

Take a square cardboard and divide it into four parts A , B , C and D as

shown

a

a

a

b

b

b

D

B

C

A


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a

a a

a

a

a

b

b

b

b D

B

C

A


11/19

D

a

b

B

b

aA

a

a

Cb

b

Area = a*a = a2 Area = a*b

Area = a*bArea = b*b = b2

Area of each portion will become as shown


12/19

( a+ b)2 = a2 + ab + ab + b2

(a+b)2 = a2 + 2ab + b2

Area of figure A = a*a = a2

Area of figure C = b*b = b2

Area of figure D = a*b

Area of figure B = a*b

On combining these


13/19

Derivation of the formula

( a-b)2 = a2 2ab + b2

Take two squares ABON and SOMR of

sides a and b as shown in the figure

a*a b*b b

b

a

a

A N

B

S

O M

R


14/19

baB O M

A

a*a b*bb

a

NS R

Area of square ABON = a*a

area of square SOMR = b*b


15/19

Taking AP = BQ = b

draw a line PQ in the square ABON which divides it in two

rectangles

ABQP and PQON of area (a*b) and [a*(a-b)]

b

a*b b*b

b

a

b (a-b)

A

B

P N

Q O

S

M

R

a

*(

a-

b)

b


16/19

Produce RS to get the following fig.

a*b b*b ba

b (a-b)

A

B

P N

Q O

S

M

R

a*(

a-b

)

b

T


17/19

Now this figure is made up of rectangle APQB, rectangle RMQT and square PNST

because

QM = (a-b) + b = aNS = NO SO = AB RM = (a-b)

therefore

(a2 + b2 ) = area (rectangle APQB) + area (rectangle RMQT) + area ( square PNST)

= ab + ab + (a-b) * (a-b)

= 2ab + (a-b)2

(a-b)2 =a2 - 2ab + b2

a*b b*b ba

b (a-b)

A

B

P N

Q O

S

M

R

a*(

a-b

)

b

T


18/19

therefore

(a2+ b2 )= area(rectangle APQB) + area(rectangleRMQT + area ( square PNST)

= ab + ab + (a-b) * (a-b)= 2ab + (a-b)2

(a-b)2 = a2- 2ab + b2


19/19

Queries

Derive the formula

(a+b+c)2=a2+b2+c2+2ab+2bc+2ca

geometrically

Can you derive more formulae

geometrically

Documents

Area of Geometric Figure