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FIROZ AHMADʼS MATHEMATICS

MATHEMATICS
Mob
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Sc
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Ed, M
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Exericies-I


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Exericies-III


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Sc
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Ed, M
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Ram Rajya More, Siwan (Bihar)

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FIROZ AHMADʼS MATHEMATICS

THINGS TO REMEMBER
 Relation between the Sides and Angles of Triangle :
In a ABC, angles are denoted by A, B and C the lengths of corresponding sides opposite to these
angles are denoted by a, b and c respectively
...

Also,
Semi-perimeter of the triangle is
s

A

c
abc
2

B
B

b
C

a

C

Sine Rule
In any ABC, the sines of the angles are proportional to the lengths of the opposite side,
ie,
sin A sin B sin C


a
b
c

It can also be written as
a
b
c


k
sin A sin B sin C

then

(say)

a = k sin A, b = k sin B, c = k sin C

Cosine Rule
In any ABC cosine of an angle can express in terms of sides
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Sc
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Ed, M
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sin

A

2

s  b s  c 

2
...


sin

C

2

s  a s  b

4
...


cos

B

2

s s  b 
ca

6
...


tan

A

2

s  b s  c 
s s  a 

8
...


tan

C

2

s  a s  b 
s s  c 

bc

ca

ab

 Area of Tringles :
In a ABC, if the sides of the tringle are a, b, c and corresponding angles are A, B, C respectively, then
area of tringle
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M
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(Mat hs), B
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Phil (M at hs)



1
ab sin C
2



1
bc sin A
2



1
ca sin B
2

Ram Rajya More, Siwan (Bihar)

3

FIROZ AHMADʼS MATHEMATICS

When One Side and Corresponding Angles are Known



c 2 sin A sin B
2 sin C



a 2 sin B sin C
2 sin A

b 2 sin C sin A

2 sin B
When all the Three Sides are Known

  ss  a s  b s  c 
It is Known as Hero’s formula
...
Circumcircle of a Triangle
The circle passing through the vertices of a ABC is called circumcircle
...
Circumcentre is the point of intersection of
perpendicular bisectors of the sides
...
R 

2 sin A 2 sin C
abc
2
...
Incircle of a Triangle

c

B

a

b
O
R

C

The circle touching all the sides of a trinangle internally is known as an incircle of the triangle
...
Incenter is the point of
intersection of besectors of the angles of the triangle
...
r 


s

A

2
...
r 
A
B
C
cos
cos
cos
2
2
2
4
...
Sc
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Ed, M
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Escribed Circles of a Triangle
The circle touching BC and the two sides AB and AC produced of ABC externally is called the
escribed circle opposite to A
...
The cente of this circle is known as excenter
and denoted by I
...


B
r1
A

O

I1

C

Similarly, r2 and r3 denote the radii of the escribed circles opposite to the angles B and C respectively and excentres are denoted by I2 and I3
...
Here,
B
C
cos
2
2
A
cos
2


A
A
B
C
 s tan  4 R sin cos cos 
1
...
r2 
sb
2
2
2
2

b cos


C
C
A
B
 s tan  4 R sin cos cos 
3
...
:
In a ABC, AD, BE and CF are perpendiculars from the vertices A, B and C respectively to the opposite
sides
...

The DEF is called the pedal triangle of ABC
1
...


PD = 2R cos B cos C, PE = 2R cos C cos A, PF = 2R cos A cos B
A

F
P

M
...
(Mat hs), B
...
Phil (M at hs)

Ram Rajya More, Siwan (Bihar)

B

D

E

C

5

FIROZ AHMADʼS MATHEMATICS

 Regular Polygon :
A regular polygon is a polygon which has all its sides as well as all its angles are equal
...

a

cos ec
2
n

Radius of circumcircle,

R

and radius of incircle,

a

r  cot
2
n

E

A
R

 nr 2 tan




n

r

B

Where, a is the length of the side of a regular polygon
...
When any three
of these six elements (except all the three angles) of a triangle are given, the triangle is known as completely
...
this
process is called the solution of triangles
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Sc
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Ed, M
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The third side is obtained either by sine
2
2
rule of cosine rule
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(iv) When two sides and angle opposite to one of them be given In this case, the triangle is not
always uniquely determined
...
So, it is called an ambiguous case
...
Now tha following cases arise
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When b < c sin B, then a is imaginary and so there is no solution
...

When

b = c sin B

then

a = c cos B,

so there is unique solution
Also,


b = c sin B
k sin C 

k sin B
sin B



sin C = 1

C = 90o
The triangle is right angled in this case
...

When
b > c sin B,
then
b2 - c2 sin2B > 0,
then there are two solutions given by
a  c cos B  b 2  c 2 sin 2 B

and
a  c cos B  b 2  c 2 sin 2 B
Now, if B is an acute angle, then there are two triangles proided that c > b > c sin B and if B is an
abtuse angle, then there is only one triangle provided that b > c
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Ed, M
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n

 If B is acute and c < b, then there is only one triangle exist
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 B is an acute angle if cos B is positive and if cos B is negative, then B is an obtuse angle
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Sc
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Ed, M

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