Area Of Geometric Figure

Geometrical representation of some formulae

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

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

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 formulae like



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



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



( a-b)2 = a2 – 2ab + b2



Lets try

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

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 EF‫ װ‬AD and GH‫װ‬AB Area of cardboard=l*b=(x+a) (x+b) D

b

x

F

a

bx

C

ab

b

(x+b) H

O

G x

ax

x

2

A

x

E

a

(x+a)

x B

Derivation of formula (x +a) (x + b) = x2 + (a+b) x + ab Take a rectangular cardboard ABCD and take (x+a) and (x+b) its sides as shown . D

C

(x+b) H

A

(x+a)

B

Derivation of formula

(x +a) (x + b) = x2 + (a+b) x + ab Divide rectangular cardboard ABCD in four portions . The area of each portion will be as shown in figure D

b

x

a

F

bx

C

ab O

b

(x+b)

G

H

x

x

ax

x2 A

x

E

a

(x+a)

B

D

F

F

C

bx

b

G

x

ab

b

O

O

a H

G

x2

x A



O

x

O x

E

E

H

ax a

B

Combine the area of each portion to get the desired result

 x2+

ax + bx + ab

(x+a)

= x2 + (a+b) x+ ab

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

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

b

B

D

C

a

b

a

b

a

a

A

b

a

B

a

a b

D a

C b

b

Area of each portion will become as shown

A

a

B

Area = a*a = a2

Area = a*b

a

D

Area = a*b a

a

b

b

C

b Area = b*b = b2 b

Area of figure A

= a*a = a2

Area of figure B =

a*b

Area of figure C = b*b = b2 Area of figure D = a*b On combining these

(

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

2

2

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

B

a A

a

O

b b*b

a*a S N

M

b R

Area of square ABON = a*a area of square SOMR = b*b B

a A

a

O

b b*b

a*a S N

M

b R

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)]

a

b

Q (a-b) O a*(a-b)

B

a*b

b*b S

A

b

P

b

N

M

b R

Produce RS to get the following fig.

a

b

Q (a-b) O a*(a-b)

B

a*b T

A

b

P

M

b*b S N

b R

a

b

Q (a-b) O a*(a-b)

B

a*b T

A

b

P

M

b*b S

b R

N

Now this figure is made up of rectangle APQB, rectangle RMQT and square PNST because QM = (a-b) + b = a NS = 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

 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

Queries  Derive

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

 Can

you derive more formulae geometrically