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# 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?

##  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

##  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 x F a C

b
b bx ab

(x+b)
G O H

x x
x2 ax

A x a B
E
(x+a)
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 B

(x+a)
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 x F a C

b
b bx
ab (x+b)
O
G H

x x
x2 ax

A x a B
E
(x+a)
D F F
C

b bx ab
b

G x O O a
H
G O O H

x x2 x ax

A x E E a B

##  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
Derivation of the formula
(a+b)2 = a2+2ab+b2
 Takea square cardboard and divide it into four
parts A , B , C and D as shown
a b

A B a

D C b

a b
a b

a A a B a

b
D C b

a b
Area of each portion will become as shown

A a
B a

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

a b

D b
C b
Area = b*b = b2
Area = a*b b
a
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
2 2 2

(a+b)2 = a2 + 2ab + b2
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 O b M

b*b b
a a*a
S R
A N
Area of square ABON = a*a
area of square SOMR = b*b

B a O b M

b*b b
a a*a
S R
A N
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 b Q (a-b) O b M

a*(a-b)
b*b b
a a*b
S R
A b P N
Produce RS to get the following fig.

B b Q (a-b) O M

b
a*(a-b)
a a*b b*b
T R
S
A b P N
B b Q (a-b) O M

a*(a-b)
a a*b b*b
T R
S
A b P 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

geometrically
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