Maths magazine

Mathematics

1. E-MATHAZINE
CLASS –X A
SESSION-2013-2014
Teacher Incharge –Ms.Sushma Singh
Group Leaders –Mansi, Chetna
Front page and Logo-Chetna

2. Click here to download this powerpoint template : Brown Floral Background Free Powerpoint Template
For more : Templates For Powerpoint
Page 2
We, the students of class 10-A
would like to thank our maths
teacher Mrs. Sushma Singhfor
her precious guidance. We would
also like to thank each and every
member of our group for their
co-operation which helped us to
complete this project successfully.

4. Advisor-Mrs.Sushma Singh
Chief Editor-Mayank Saini
Editor-
*Gaurav Dabas
* Mansi Manwal
*Chetna
*Shivam Arora
*Diksha
*Himanshi
*Sandeep
*Nishant
*Kamaljeet
*Yasmin
*Priyanka

5. WORK BREAKDOWN
ROLLNO.NAME WORKGIVEN
1 Akram Story and News
2 Alisha Raptiles and saree designs
3 Anuradha Painting and Geometrical designs
4 Ashish Cartoons and paintings
5 Chetna Work breakdown and limitations
6 Deepika Logical reasoning and series
7 Deeksha Essential questions
8 Gaurav Mental maths Q/A and activity
9 Gaurav Dabas Conclusion and product
10 Himanshi Introduction and lessons
11 Kamaljeet Analysis and data interpretation

6. ROLL NO. NAME WORK GIVEN
12 Kaustubh Geometrical designs and quiz
13 Mansi Manwal Sugesstions and bibilography
14 Mayank Saini Historical background
15 Md. Shahid Rza Story and puzzle
16 Naveen Activity and small project
17 Naveen Singh Cartoons and game
18 Nishant Methodology
19 Nitin Bisht Mathematician and article
20 Nitish Mathematician and graphs
21 Payal Arora Aims and objective
22 Pooja Graphs and article
23 Prabhat Article and news
24 Pragati Story and puzzle

7. ROLL NO. NAME WORK GIVEN
25 Preeti Sudha Series and mathematician
26 Priyanka Abstract of the project
27 Rajat Makkar Geometrical designs and topics
28 Rupam Activity and article
29 Sagar Rana Puzzles and news
30 Sandeep Reference
31 Sahil Dash Graphs and painting
32 Shivam Arora Tools used
33 Shivam Gulati Mental maths Q/A and
reasoning
34 Sumit Quiz and viva
35 Yasmin Topics and presentation

8. Our project is in a form ofmagazine. We all give our 100% to complete this
project. We faced many problems butSushma Singhmamhelp usa lotto
make this project successfully.Thismagazinedescribe topics like:-story,
cartoon, mathematics, games, article, newspaper, group activity,
geometrical design,hots, reptiles, mathematicaldesigns and many other
topics. You can look up all topics and get knowledge about them by
thismagazine. Our whole class did a tremendous job. The work was
distributed by the chief editor, editor, and our teacher
Mrs.SushmaSinghMam. We think that
ourmagazineis easilyunderstandablebecause we used easylanguage
and someinterestingfacts so you don't loses your interest. Everyone gives
his and her all effort to the project to make it awesome,
easilyunderstandableand as well asinteresting also. Such that everyone
want to read thismagazine either he/she be a teacher or student or even
pass out from college.We hope that all of you will like thismagazine andall
the teachers and student who read get a vast knowledge from
thismagazine.

9. Click here to download this powerpoint template : Brown Floral Background Free Powerpoint Template
For more : Templates For Powerpoint
Page 9
Mathematicsfirstarosefromthepractical
needtomeasuretimeandtocount.The
earliestevidenceofprimitiveformsof
countingoccursinnotchedbonesand
scoredpiecesofwoodandstone.Early
usesofgeometryarerevealedinpatterns
foundonancientcavewallsandpottery.
AscivilisationsaroseinAsiaandtheNear
East,sophisticatednumbersystemsand
basicknowledgeofarithmetic,geometry,
andalgebrabegantodevelop.

10. Click here to download this powerpoint template : Brown Floral Background Free Powerpoint Template
For more : Templates For Powerpoint
Page 10
For more than two thousand years, mathematics has been a part of
the human search for understanding. Mathematical discoveries
have come both from the attempt to describe the natural world and
from the desire to arrive at a form of inescapable truth from careful
reasoning. These remain fruitful and important motivations for
mathematical thinking, but in the last century mathematics has
been successfully applied to many other aspects of the human
world: voting trends in politics, the dating of ancient artifacts, the
analysis of automobile traffic patterns, and long-term strategies for
the sustainable harvest of deciduous forests, to mention a few.
Today, mathematics as a mode of thought and expression is more
valuable than ever before. Learning to think in mathematical terms
is an essential part of becoming a liberally educated person.

12. Click here to download this powerpoint template : Brown Floral Background Free Powerpoint Template
For more : Templates For Powerpoint
Page 12
REFRENCES

13. Indian mathematicians have made a
number of contributions to mathematics
that have significantly influenced
scientists and mathematicians in the
modern era. These include place-value
arithmetical notation, the ruler, the
concept of zero, and most importantly,
the Arabic-Hindu numerals
predominantly used today.

14. content

15. Click here to download this powerpoint template : Brown Floral Background Free Powerpoint Template
For more : Templates For Powerpoint
Page 15
ARYABHATTA

16. Aryabhata was born in Taregna, which is a small
town in Bihar, India, about 30km from Patna
(then known as Pataliputra), the capital city of
Bihar State. Evidences justify his birth there. In
Taregna Aryabhata set up an Astronomical
Observatory in the Sun Temple 6th century.
There is no evidence that he was born outside
Patliputra and traveled to Magadha, the centre of
instruction, culture and knowledge for his studies
where he even set up a coaching institute. However,
early Buddhisttexts describe Ashmakas as being
further south, in dakshinapathor theDeccan, while
other texts describe the Ashmakas as having
foughtAlexander.

17. It is fairly certain that, at some point, he went
to Kusumapura for advanced studies and that he
lived there for some time. A verse mentions
that Aryabhatta was the head of an institution
at Kusumapura, and, because the university
ofNalandawas in Patliputra at the time and
had an astronomical observatory, it is
speculated that Aryabhata might have been the
head of the Nalanda university as well.
Aryabhata is also reputed to have set up an
observatory at the Sun temple inTaregana,
Bihar.

18. ARYABHATTA’S
Contribution
in math’s
Place value
system
Trigonometry
Algebra
Approximation
of pie
Numeration
Indeterminate
equation

19. The Aryabhatta numerationis a system of
numerals based on Sanskrit phonemes. It
was introduced in the early 6th century by
Āryabhaṭa, in the first chapter titled Gītika
Padamof his Aryabhatiya. It attributes a
numerical value to each syllable of the
form consonant vowel possible in Sanskrit
phonology, from ka= 1 up to hau= 10

21. and

22. In Ganitapada 6, Aryabhata gives the area of a
triangle as
tribhujasya phalashariram samadalakoti
bhujardhasamvargah
that translates to: for a triangle, the result of a
perpendicular with the half-side is the area.
Aryabhata discussed the concept of sine in his
work by the name of ardha-jya. Literally, it
means "half-chord". For simplicity, people started
calling it jya. When Arabic writers translated his
works from Sanskrit into Arabic, they referred it
as jiba

23. Theplace-value system, first seen in the 3rd
centuryBakhshali Manuscript, was clearly in place in his work.
While he did not use a symbol forzero, the French
mathematicianGeorges Ifrahexplains that knowledge of zero
was implicit in Aryabhata'splace-value systemas a place holder
for the powers of ten withnullcoefficients
However, Aryabhata did not use the Brahmi numerals. Continuing
theSanskrittradition fromVedic times, he used letters of the
alphabet to denote numbers, expressing quantities, such as the
table of sine's in amnemonicform.

24. Click here to download this powerpoint template : Brown Floral Background Free Powerpoint Template
For more : Templates For Powerpoint
Page 24

25. Srinavasaramanujanisoneofthe
celebratedIndianmathematician.His
importantcontributiontothefield
includeshardy-ramanujanlittlewood
circle,methodinnumbertheory,rode
v-ramanujan’sidentitiesinthepartition
innumbers,workonalgebraic
inequalities,ellipticfunction,continued
faction,partialsumsandproductof
helpergeometricseries.etc.andmany
more.
Srinivasaramanujanborn on22
nd
December1887inmadras,India.He
borninmiddleverylittlepoorfamily.
Likesophismgasmanhereceivedno
formaleducationinmathematicsbut
madeimportantcontributiontothe
advancementofmathematics.Hegave
usmanyformulasandtoalloverthe
worldbywhichweareabletodayto
inventmoderntechnologies

27. Click here to download this powerpoint template : Brown Floral Background Free Powerpoint Template
For more : Templates For Powerpoint
Page 27
Archimedes

28. Archimedes was born in Syracuse,
Greece in 287 BC and died 212
BC . He was the son of an
astronomer : Phidias. Archimedes
received his formal education in
Alexandria, Egypt which at the
time was considered to be the
'intellectual center' of the world.
When he completed his formal
studies in Alexandria, he returned
and stayed in Syracuse for the
rest of his life

29. Discovered
how to find
the volume
of a sphere
and
determined
the exact
value of Pi.

30. It is believed that
he was actually
the first to have
invented integral
calculus, 2000
years before
Newton and
Leibniz

31. Powers of Ten,
a way of
counting that
refers to the
number of 0's
in a number
which
eliminated the
use of the
Greek alphabet
in the
counting
system

32. A formula to
find the area
under a curve,
the amount of
space that is
enclosed by a
curve.

33. Click here to download this powerpoint template : Brown Floral Background Free Powerpoint Template
For more : Templates For Powerpoint
Page 33
Pythagoras

35. The Pythagoreans' believed “All is Number,”
meaning that everything in the universe depended
on numbers. They were also the first to teach that
the Earth is a Sphere revolving around the sun.
Pythagoras is often
considered the first
true
mathematician.

36. Pythagoras was born on Samos a Greek
island off the coast of Asia Minor. He
was born to Pythais (mom) and
Mnesarchus (dad).
As a young man, he left his native city for
Southern Italy, to escape the tyrannical
government. Pythagoras then headed to
Memphis in Egypt to study with the priests
there who were renowned for their wisdom. It
may have been in Egypt where he learned
some geometric principles which eventually
inspired his formulation of the theorem that
is now called by his name.
Towards the end of his life he fled to Metapontum because
of a plot against him and his followers by a noble of Croton
named Cylon. He died in Metapontum around 90 years old
from unknown causes.
Life

37. Many of Pythagoras’
beliefs reflect those of
the Egyptians. The
Egyptian priests were
very secretive. The
refusal to eat beans or
wear animal skins and
striving for purity were
also characteristics of
the Egyptians.

38. Click here to download this powerpoint template : Brown Floral Background Free Powerpoint Template
For more : Templates For Powerpoint
Page 38
a
2
+ b
2
= c
2

39. Proof of Pythagorean theorem by rearrangement of 4
identical right triangles. Since the total area and the areas of
the triangles are all constant, the total black area is constant.
But this can be divided into squares delineated by the
triangle sides a, b, c, demonstrating that a
2
+b
2
= c
2
.

40. Click here to download this powerpoint template : Brown Floral Background Free Powerpoint Template
For more : Templates For Powerpoint
Page 40
The sum of the angles of a triangle is equal to two right
angles or 180 degrees
The five regular solids
Venus as an evening star was the same planet
as Venus as a morning star.
The abstract quantity of
numbers. There is a big step
from 2 ships + 2 ships = 4
ships, to the abstract result 2
+ 2 = 4

41. Article

42. Mathematicsis one kind of science. We cannot do a single moment
without mathematics.Mathematics has made our everyday life easy
and comfortable.In official and personal life
become paralyzed without mathematics.
Different kinds of functions are performed by
mathematics.Mathematics works with numbers, counting, and
numericaloperations. It is used to calculate something. It can be
done both technically and manually. Large andcomplicated
mathematical problemsis solved by the computer software. On the
other hand, easymathematical operationsare performed without
the help of any machine or computer software. In official works like
banking, policy, school, college and universities mathematical
calculations are done by technically.

43. •The merits of mathematics in our everyday lifecannot be
described in words. It has opened a new dimension to us.We
cannot do a single day without mathematics.Mathematics helps us
to solvedifficult mathematical problems.It has enriched our
life.Mathematicshelps us to decide if something is a good, risky or
not.Mathematicshelps us to create everything as without the
application ofmathematics.We cannot create any building,
picture, furniture, good art, wallpaper, your room, bridge etc. It
shows us to become beneficial in life.
.
•In the end, it can be said thatmathematics helps us to take any
kinds of decision.It works just like a mentor.Without
mathematics, we never take any decision.Our everyday life
depends much on mathematics.We cannot go even an inch without
mathematics.Our everyday life becomes paralyzed without
mathematics. Therefore, it can be said thatmathematics is a part
andparcelin our everyday life.

47. Graphs

48. Click here to download this powerpoint template : Brown Floral Background Free Powerpoint Template
For more : Templates For Powerpoint
Page 48

49. Click here to download this powerpoint template : Brown Floral Background Free Powerpoint Template
For more : Templates For Powerpoint
Page 49
TYPES OF GRAPH
Bar graph
Double bar graph
Histogram
Frequency polygon
Equation graph
More than Ogive
Less than ogive

50. Click here to download this powerpoint template : Brown Floral Background Free Powerpoint Template
For more : Templates For Powerpoint
Page 50

51. Click here to download this powerpoint template : Brown Floral Background Free Powerpoint Template
For more : Templates For Powerpoint
Page 51

52. Click here to download this powerpoint template : Brown Floral Background Free Powerpoint Template
For more : Templates For Powerpoint
Page 52

53. Click here to download this powerpoint template : Brown Floral Background Free Powerpoint Template
For more : Templates For Powerpoint
Page 53

54. Click here to download this powerpoint template : Brown Floral Background Free Powerpoint Template
For more : Templates For Powerpoint
Page 54

55. Click here to download this powerpoint template : Brown Floral Background Free Powerpoint Template
For more : Templates For Powerpoint
Page 55

56. Click here to download this powerpoint template : Brown Floral Background Free Powerpoint Template
For more : Templates For Powerpoint
Page 56

57. Click here to download this powerpoint template : Brown Floral Background Free Powerpoint Template
For more : Templates For Powerpoint
Page 57

58. Click here to download this powerpoint template : Brown Floral Background Free Powerpoint Template
For more : Templates For Powerpoint
Page 58

59. Click here to download this powerpoint template : Brown Floral Background Free Powerpoint Template
For more : Templates For Powerpoint
Page 59

60. Click here to download this powerpoint template : Brown Floral Background Free Powerpoint Template
For more : Templates For Powerpoint
Page 60

61. Click here to download this powerpoint template : Brown Floral Background Free Powerpoint Template
For more : Templates For Powerpoint
Page 61

62. Click here to download this powerpoint template : Brown Floral Background Free Powerpoint Template
For more : Templates For Powerpoint
Page 62

63. shapes

64. Regular Solids
Tetrahedron
Cube
Octahedron
Dodecahedron
Icosahedron

65. triangles

66. IN FLAGS

67. IN NATURE

68. Click here to download this powerpoint template : Brown Floral Background Free Powerpoint Template
For more : Templates For Powerpoint
Page 68
Circles

70. IN MUSIC

74. IN BUILDINGS

75. Click here to download this powerpoint template : Brown Floral Background Free Powerpoint Template
For more : Templates For Powerpoint
Page 75

76. INTRODUCTION
Markthreenon-collinear
pointP,QandRonapaper.
Jointhesepointsinall
possibleways. The
segmentsarePQ,QRandRP.
Asimpleclosecurveformed
bythesethreesegmentsis
calledatriangle.Itisnamed
inoneofthefollowingways.
TrianglePQR,TrianglePRQ,
TriangleQRP,TriangleRPQ
orTriangleRQP

77. Triangle
Atriangleisapolygonofthreesides.Infact,itisthepolygonwiththeleast
numberofsides.AtrianglePQRconsistsofallthepointsontheline
segmentPQ,QRandRP.
Sides:Thethreelinesegments,PQ,QRandRPthatformthetrianglePQ,are
calledthesidesofthetrianglePQR.
Angles:Atrianglehasthreeangles.Infigure4-1,thethreeanglesare∠PQR,
∠QRPand∠RPQ.
Partsoftriangle:Atrianglehassixparts,namely,threesides,PQ,QR
andRP,andthreeangles∠PQR,∠QRPand∠RPQ.Thesearealsoknownas
theelementsofatriangle.
VerticesofaTriangle
Thepointofintersectionofthesidesofatriangleisknownasitsvertex.In
figure4-1,thethreeverticesareP,QandR.Inatriangle,anangleisformed
atthevertex.Sinceithasthreevertices,sothreeanglesareformed.The
wordtriangle=tri+angle‘tri’meansthree.So,trianglemeansclosedfigure
ofstraightlineshavingthreeangles.

78. Classification of Triangles :
Triangles can be classified in two groups
A.Trianglesdifferentiated
onthebasisoftheir
sides.
1.EquilateralTriangle: A
trianglewithallsidesequalto
oneanotheriscalledEquilateral
Triangle.Here,PQ=QR=RP.
ThereforeTrianglePQRisan
EquilateralTriangle.

79. 2.IsoscelesTriangle:Atriangle
withapairofequalsidesiscalled
anIsoscelesTriangle.
Here,PQ=QR.
ThereforeTrianglePQRisan
IsoscelesTriangle.

80. 3.ScaleneTriangle:Atrianglein
whichallthesidesareofdifferent
lengthsandnotwosidesareequal,
thetriangleiscalledascalene
triangle.
HerePQ≠QR≠PR.
ThereforeTrianglePQRisan
ScaleneTriangle.

81. B. Triangles differentiated
on the basis of their angles.
1. Acute angled triangle: A
triangle whose all angles are acute
is called an acute-angled triangle or
simply an acute triangle.
Or
If all the three angles of a triangle
are less than 90 degree then it is an
Acute Angled Triangle.

82. 2.Right-angledtriangle:A
trianglewhoseoneofthe
anglesisarightangleiscalleda
right-angledtriangle,orsimply
arighttriangle.
Thesideoppositetotheright
angleiscalledHypotenuseand
theothertwosidesarecalled
thelegsofthetriangle.Ina
righttriangle,hypotenuseisthe
greatestside.
Or
Ifoneangleofatriangleis
equalto90degreethenitisa
RightAngledTriangle.

83. 3. Obtuse-angled triangle: A
Triangle one of whose angle is
Obtuse is called an Obtuse-
angled triangle or simply an
Obtuse Triangle.
Or
If one angle of a triangle is
greater then 90 degree then it is a
Obtuse Angled Triangle.

84. Properties of a Triangle
1.AngleSumProperty:Thesumofthe
measuresofthethreeanglesofatriangleis180.
Letusprovethisproperty
Proof:LetPQRbeanytriangle.DrawalineAB
paralleltoQR,passingthroughP.Markthe
anglesasshowninfigure4-13.
Now, ∠4 = ∠2 (Alternate angles)
∠5= ∠3 (Alternate angles)
Therefore
∠1+ ∠2 + ∠3= ∠1+ ∠4+ ∠5= 180
(Linearity property)
Therefore, Sum of the angles of a triangle is 180
degree (Two right angles). Hence proved

85. 2. ExteriorAngleProperty:In a triangle, the
measure of an exterior angle equals the sum of
the measures of the remote interior angles.
In figure 4-14, Triangle PQRis shown. The
side QR is extended to S. Now, ∠PRSis the
exterior angle. With reference to the exterior
∠PRS, ∠PQRand ∠RPQare remote interior
angles (oropposite interior angles). Let us
prove thisproperty
Proof:Refer to figure 4-14
∠PRSor ∠4 is an exterior angle. We have to
prove that ∠4= ∠1+ ∠2
We know that the sum of the anglesof a
triangle equals 180 degree
In Triangle PQR,∠1+ ∠2 + ∠3= 180 … (1)
Since ∠3 and ∠4 form a linear pair ∠3 + ∠4=
180 … (2)
From (1) and (2), we have
∠3+ ∠4=∠1 + ∠2+ ∠3 or
∠4= ∠1+ ∠2 Hence Proved
Since ∠4= ∠1+ ∠2, this implies that ∠4> ∠1
and ∠4> ∠2, Therefore, the exterior angle of a
triangle isgreater than each remote interior
angle.

86. 3. Triangles Inequality:The sum of the lengths of any two sides of
a triangle is greater than the length of the thirdside.
Let us prove thisproperty Here, we have to prove that PQ+ PR>
QR, PQ+ QR> PRand QR+ PR> PQ.
Extend side QP to S such thatPS=PR. Join SR.
Proof:In Triangle PQR, we have PS=PR (By construction)
Thus Triangle PSR is an isosceles triangle. We know that in an
isosceles triangle, angles opposite equal sides are equal
Therefore∠PSR= ∠PRS… (1)
Or ∠PRS= ∠PSR(Identity congruence)
∠QRP+ ∠PRS> ∠PRS (The sum of two non-zero numbers is
greater than each individual number)∠QRP + ∠PRS > ∠PSR
(From 1)
Or ∠QRS> ∠PSR(Since∠QRPand∠PRSare adjacent angles)
Or ∠QRS > ∠QSR (Since∠PRSand∠QSR is the same angle)
or QS > QR (Since, side opposite to greater angle is greater)
or QP+PS > QR (SinceQS=QP+PS)
or QP+PS > QR (SincePS=PR by construction)
or PQ+PR > QR (Since QP and PQ the same side of the triangle
taken in different order)

87. Congruence of TrianglesXYZABC
Definition :Two triangles are congruent if three sides and three angles of
one triangle are equal to the corresponding sides and angles of other triangle.
The congruence of two triangles follows immediately from the congruence
of three lines segments and three angles.
Given two triangles ΔABC and ΔXYZ.
If AB is congruent to XY, Ais congruent to X,
BCis congruent to YZ, Bis congruent to Y ,
CAis congruent to ZX, Cis congruent to Z
then we say that ΔABC is congruent to ΔXYZ, and we write
X
Y
Z
A
B
C

88. 1. Side-Angle-Side Principle
Given two triangles ΔABC and ΔXYZ.
If AB is congruent to XY
Bis congruent to Y
BCis congruent to YZ
then ΔABC is congruent to ΔXYZ
A
B
C
X
Y
Z

89. 2. Angle-Side-Angle Principle
Given two triangles ΔABC and ΔXYZ.
If A is congruent to X
AC is congruent to XZ
C is congruent to Z
then ΔABC is congruent to ΔXYZ
A
B
C
X
Y
Z
((

90. 3. Side-Side-Side Principle
Given two triangles ΔABC and ΔXYZ.
If AB is congruent to XY
BC is congruent to YZ
CA is congruent to ZX
then ΔABC is congruent to ΔXYZ
A
B
C
X
Y
Z

91. Theorem
If ΔABC is congruent to ΔXYZ ,
then
AB is congruent to XY
BC is congruent to YZ
CA is congruent to ZX
and
A is congruent to X
B is congruent to Y
C is congruent to Z
In short, corresponding parts of
congruent triangles are congruent.

92. Example 14.5
Show that the diagonals in a kite is perpendicular to each other.
Recall that a kite is a quadrilateral with 2 pairs of
congruent adjacent sides. In particular for the following
figure, AB = ADand CB = CD.
A
BD
C
E

93. A
BD
C
We first need to show that ΔADC and ΔABC are congruent.
This is true because
AD = AB
DC = BC
AC = AC
and we have the SSS
congruence principle.
Therefore, (click)
12
1 is congruent to 2
(b/c it is a kite)
(b/c it is a kite)
(b/c they are the same side)

94. A
BD
C
E
12
Now we only considerΔADEand ΔABE.
They should be congruent because
AD = AB
1 = 2
AE = AE
Hence SAS principle applies.
AED is congruent to AEB, and they both add up
to 180, hence each one is 90.

95. Similarity of Triangles
Definition
Given ΔABC and ΔXYZ.
If A is congruent to X
B is congruent to Y
C is congruent to Z
and AB :XY = BC :YZ = CA :ZX
then we say that ΔABC is similar to ΔXYZ, and the
notation is
ΔABC ~ ΔXYZ
A
B
C
X
Y
Z

96. 1. SSS similarity principle
Given ΔABC and ΔXYZ.
If AB :XY = BC :YZ = CA :ZX
then ΔABC is similar to ΔXYZ.
A
B
C

97. 2. AAA similarity principle
Given ΔABC and ΔXYZ.
If A is congruent to X
B is congruent to Y
C is congruent to Z
then ΔABC is similar to ΔXYZ
C
X
Y
ZA
B

98. 3. AA similarity principle
Given ΔABC and ΔXYZ.
If A is congruent to X
B is congruent to Y
then ΔABC is similar to ΔXYZ
(because the angle sum of a triangle is always
180
o
)
A
B
C
X
Y
Z

99. 4. SAS similarity principle
Given ΔABC and ΔXYZ.
If AB :XY = BC :YZ and
B is congruent to Y
then ΔABC is similar to ΔXYZ
A
B
C
X
Y
Z

100. Some More Properties of Triangle
1.The angles opposite to equal
sides are always equal.
Example: In figure
Given: ▲ABC is an isosceles triangle in which
AB = AC
TO PROVE: ∠B = ∠C
CONSTRUCTION : Draw AD bisector of ∠BAC
which meets BC at D
PROOF: IN ▲ABD & ▲ACD
AB = AC (GIVEN)
∠BAD = ∠CAD (GIVEN)
AD = AD (COMMON)
▲ABD and ▲ACD are similar triangles (BY
SAS RULE)
Therefore, ∠B = ∠C
A
B C
D

101. 2. The sides opposite to equal angles
of a triangle are always equal.
Example : In Figure
Given : ▲ABC is an Isosceles triangle in which
∠B = ∠C
TO PROVE: AB = AC
CONSTRUCTION : Draw AD the bisector of
BAC which meetsBC at D
Proof : IN ▲ABD & ▲ACD
∠B = ∠C (GIVEN)
AD = AD (GIVEN)
∠BAD = ∠CAD (GIVEN)
Therefore, ▲ABD &▲ACD are similar
triangles (BY ASA RULE)
Therefore, AB = AC
A
B C
D

102. Inequality
When two quantities are unequal then on comparing these
quantities we obtain a relation between their measures called
Inequalityrelation.
THEOREM 1. If two sides of a triangle are unequal the larger side
has the greater angle opposite to it.
Given : IN ▲ABC , AB >AC
TO PROVE : ∠C = ∠B
Draw a line segment CD from vertex such that AC = AD
Proof : IN ▲ACD , AC = AD
∠ACD = ∠ADC ---(1)
But ADC is an exterior angle of ▲BDC
∠ADC > ∠B ---(2)
From (1) & (2)
∠ACD > ∠B ---(3)
∠ACB > ∠ACD ---4
From (3) & (4)
∠ACB > ∠ACD > ∠B , ∠ACB > ∠B ,
Therefore, ∠C > ∠B
A
B C
D

103. THEOREM 2.In a triangle the greater angle has a large
side opposite to it
Given: IN ▲ABC ∠B > ∠C
TO PROVE : AC > AB
PROOF : We have the three possibility for sides AB and AC
of ▲ABC
(i)AC = AB
If AC = AB then opposite angles of the equal sides are
equal than
∠B = ∠C
But we know AC ≠ AB
(ii) If AC < AB
We know that larger side has greater angles opposite to
it.
AC < AB , ∠C > ∠B, we know that AC is not greater then
AB
(iii)If AC > AB
We have left only this possibility
AC > AB
A
CB

104. THEOREM 3.The sum of any two angles is
greater than its third side
TO PROVE : AB + BC > AC
BC + AC > AB
AC + AB > BC
CONSTRUCTION: Produce BA to D such that AD
+ AC .
Proof: AD = AC (GIVEN)
∠ACD = ∠ADC (Angles opposite to equal sides
are equal ) ---(1)
∠BCD > ∠ACD ----(2)
From (1) & (2)
∠BCD > ∠ADC = BDC
BD > AC (Greater angles have larger opposite
sides )
BA + AD > BC (BD = BA + AD)
BA + AC > BC (By construction)
Therefore, AB + BC > AC and BC + AC >AB
CB
D
A

105. THEOREM 4.Of all the line segments
that can be drawn to a given line from
an external point , the perpendicular
line segment is the shortest.
Given : A line AB and an external
point.
Join CD and draw CE perpendicular to
AB
TO PROVE CE < CD
PROOF : IN ▲CED, ∠CED = 90
0
THEN ∠CDE < ∠CED
CD < CE ( Greater angles have larger
side opposite to them. )
BA
C
ED

106. 1.If the altitude from one vertex of a triangle
bisects the opposite side, then the triangle
is isosceles triangle.
Given : A ▲ABC such that the altitude AD from
A on the opposite side BC bisects BC I. e. BD =
DC
To prove : AB = AC
SOLUTION : IN ▲ADB & ▲ADC
BD = DC
∠ADB = ∠ADC = 90
AD = AD (COMMON )
Therefore, ▲ADB & ▲ADC are similar triangles
(BY SAS RULE )
Hence, AB = AC
A
CDB

107. 2. In a isosceles triangle altitude from the vertex
bisects the base .
EXAMPLE: (in fig. 2.6)
GIVEN: An isosceles triangle AB = AC
To prove : D bisects BC i.e. BD = DC
Proof: IN ▲ADB & ▲ADC
∠ADB = ∠ADC
AD = AD
∠B = ∠C ( Given: AB = AC)
Therefore, ▲ADB & ▲ADC are similar triangles (By ASA)
Hence, BD = DC (BY CPCT)
A
CDB

108. 3. If the bisector of the vertical angle of a triangle bisects the base of the
triangle, then the triangle is isosceles.
GIVEN: A ▲ABC in which AD bisects ∠A meeting BC in D such that BD = DC,
AD = DE
To prove : ▲ABC is isosceles triangle .
Proof: In ▲ADB & ▲EDC
BD = DC
AD = DE
∠ADB = ∠EDC
Therefore, ▲ADB & ▲EDC are similar triangles (By SSA)
Therefore, AB = EC
∠BAD = ∠CED (BY CPCT)
∠BAD = ∠CAD (GIVEN)
Hence, ∠CAD = ∠CED
And AC = EC (SIDES OPPOSITE TO EQUAL ANGLES ARE EQUAL)
AC = AB , HENCE ▲ABC IS AN ISOSCELES TRIANGLE.
E
D
C
B
A

109. Games

110. MATHEMATICAL GAMES
Times hitori……
1 2 3 6 6 8 2 9
1 4 3 5 6 9 2 8
8 8 2 8 9 1 2 3
5 5 4 8 3 5 7 9
2 3 2 7 2 4 9 6
1 9 1 4 2 8 3 5
5 1 3 2 6 6 5 7
9 6 8 3 1 7 5 4
How to play
(1) A number may repeat just once in each row or
column. Eliminate repeating number by darkening
cells.

111. (2)Darkened cells must never be adjacent in a row or
column.
(3)Unmarked cells must create a single continuous
area undivided by darkened cells.
(4)
Every time you darken a cell you can
automatically circle its vertical and
horizontal neighbours which
may cannot be eliminated.

112. Any cell ‘SEND WICHED’
between neighbours of the
some value can be
circled.
(5)

113. Example:
8 55 7 3 54 6
4 5 1 67 6 52
2 57 1 5 4 6 8
5 6 2 4 37 63
2 7 55 1 72 7
1 6 6 6 4 3 5 7
7 8 4 6 2 3 3 7
8 4 5 2 35 7 1
(6)A ‘triple’ is a special case of sand which circle the
centre cell and darken the ends. e.g. 666.

114. cartoons
Hello Mickey. How
are you?
I am fine Ben.
Mickey do you know
about real numbers?
Yes I know in class
Xth. I had read about
real numbers.

115. Can you
describe me
about this?
Yes why not?
Thanks.
First you should know
what are two very
important properties of
real numbers.

116. What are these
two properties ?
(1) Euclid’s division lemma
(2)Fundamental theorem
of arithmetic.
What are the main
role of these
properties?
Let 2 positive integers a
and b. There exist unique
integers q and r satisfying
a=bq+r,0<r<b

117. Step (1) We find the whole no. q and r such that c=dq+r,
0<r<d.
Step (2) If r=0,d is the HCF of c and d . If r is not= 0, apply
division lemma to d and r.
Step (3) Continue the process till the remainder is 0.
For example: HCF of 4052 and 12576
Here, 12576=4052.8+420
4052=420 .9+272
420=272.1+148
272=148.1+124
148=24.5+4
24=4.6+0
So, 4 is the HCF of 12576 and 4052.

118. According to this theorem , every
composite number can be expressed as a
product of prime and this factorization is
unique a par from the order in which
prime factors occur.
For example, 32760=2.2.2.3.3.5.13
It is very easy. Now tell me
about Fundamental theorem
of Arithmetic.
Yo! I understand very
formly about real numbers.
Thanks.

119. Click here to download this powerpoint template : Brown Floral Background Free Powerpoint Template
For more : Templates For Powerpoint
Page 119
Mathematical series is a sequence or correct
ordering of mathematical figures or numbers
according to there pattern.
π
+
=
-

120. Click here to download this powerpoint template : Brown Floral Background Free Powerpoint Template
For more : Templates For Powerpoint
Page 120
Basic concepts
Here we work at mathematical series.
As we know mathematical series is a
sequence or correct ordering of
mathematics figures or numbers by
according to there pattern. Here we
use two types of mythical series
•Mathematical drawing series
•Number series.

121. Click here to download this powerpoint template : Brown Floral Background Free Powerpoint Template
For more : Templates For Powerpoint
Page 121
Mathematical Drawing Series
?
?
-
-

122. Click here to download this powerpoint template : Brown Floral Background Free Powerpoint Template
For more : Templates For Powerpoint
Page 122
-
-
?
?
-?

123. Click here to download this powerpoint template : Brown Floral Background Free Powerpoint Template
For more : Templates For Powerpoint
Page 123
Number series
4 , 9 , 16 , 25 ,-?
1 , 3 , 6 , 10 , -?
8 ,27,64,125, -?

124. Activities

125. •Objective
•To verify using the method of paper cutting, pasting and folding
that the lengths of tangents drawn from an external point are
equal.
•Pre-requisite knowledge
•Meaning of tangent to a circle.
•Materials required
•coloured papers,
•pair of scissors,
•ruler,
•sketch pens,
•compass,
•pencil.
To verify that length of tangents drawn from an
external point to a circle are equal

126. •1. Draw a circle of any radius on a coloured paper and cut it. Let
O be its centre.
•2. Paste the cutout on a rectangular sheet of paper.[Fig 10(a)]
•3. Take any point P outside the circle.
•4. From P fold the paper in such a way that it just touches the
circle to get a tangent
•PA (A is the point of contact). [Fig 10(b)]. Join PA.
•5. Repeat step 4 to get another tangent PB to the circle (B is the
point of contact).
•[Fig 10(c)]. Join PB.
•6. Join the centre of the circle O to P, A and B. [Fig 10(d & e)]
•7. Fold the paper along OP. [Fig 10(f)] What do you observe?
•Observations
•Students will observe that
•1. Δ OPA and Δ OPB completely cover each other.
•2. Length of tangent PA = Length of tangent PB.

127. Designs

131. Click here to download this powerpoint template : Brown Floral Background Free Powerpoint Template
For more : Templates For Powerpoint
Page 131

132. Mentalmaths questions
1) Express 1000 as a product of prime factor ?
➢23x 53= 1000.
2) Which prime numbers will be repeatedly
multiplied in prime factorizatation of 3200 ?
➢2.
3) Find the digit at units place of 8nif n is a
multiple of five ?
➢8.
4) What are the prime factors of denominators
of fraction 7/80 ?
➢7/24x 5

133. 5)If HCF of two no is 68 & 85. what is the
LCM of two numbers ?
➢340.
6)What is the HCF of 95 & 152 ?
➢14.
7)Find the no which when divided by 18 gives
the quotient and reminder as 7 & 4?
➢130.
8)176 when divided by a no gives the
reminder 5 & quotient 9 ? What is the
no?
➢19.

134. 9)By which smallest irrational number±27 be
multiplied so as to get a rational number ?
➢should be multiply by ±3 to get a rational no.
10) What is the product of (±7 + ±5) and (±7-±5)
?
➢2.

135. QUIZ
Ques1.A number consists of two digits whose sum is
5,When the digits are reversed, the number becomes
greater by 9. Find the number?
Answer: 23
Ques2.The diameter of a circle whose area is equal to
the sum of the areas of the two circles of radii 40cm
and 9cm is:
Answer: 82cm

136. Ques3.The number of cubes of side 2cm which can
be cut from a cube of side 6cm is:
Answer:27
Ques4.The value of K for which the pair of
equations K x-y=2 and 6x-2y=3 has a unique
solution.
Answer; k is not equal to 3

137. Ques5.In a triangle right angle at Q .In which
PR=13cm and PQ=12cm,find tan P –cot
R.
Answer; tan P –cot R=0
Ques6.If sin A=3/5,calculate Cos A.
Answer; Cos A=4/5.

138. Ques7.The value of tan1,tan2,tan3….tan89 is;
Answer;1
Ques8.Draw a figure in which triangle ACB is
similar to triangle APQ .If BC=8cm,
PQ=4cm,BA=6.5cm,AP=2.8cm,find CA and
AQ.
Answer; AC=5.6cm,AQ=3.25cm.

139. Ques9.Draw a triangle in which d is the mid point
of side AB and e is the mid point of side AC
AD=3cm,BD=4cm and DE=2cm ,if DE is
parallel to BC then x is equal to:
Answer;4.7cm
Ques10.(sec A +tan A)(1-sinA)
Answer: cos A

140. Ques11.All circles are ;
Answer; Similar
Ques12.Half the perimeter of a rectangular garden
,whose length is 4cm more than its width , is
36m.Find the dimensions of the garden.
Answer; Length=11m and width=7m

141. VIVA Questions
Ques1.If a+1,2a+1,4a-1 are in A.P then value of a is:
(A)1 (B)2
(C)3 (D)4
Answer;(B)2
Ques2.If the 3
rd
and 9
th
terms of an AP are 4 and -8 respectively ,which term
of this AP is 0?
(A)4
th
(B)5
th
(C)6
th
(D)7
th
Answer;(B)5
th

142. Ques3.The lengths of the diagonals of a rhombus are 24cm,and 32cm The
perimeter of the rhombus is O?
(A)9cm (B)128cm
(C)80cm (D56cm
Answer; (C)80cm
Ques4.What is the value of sin A at the thirty degree angle;
(A)0 (B)1/2
(C)1 (D)2
Answer: (B)1/2

143. Ques5.If the diameter of a protractor is 7cm then its perimeter is:
(A)18cm (B)20cm
(C)22cm (D)26cm
Answer: (A)18cm
Ques6.If the surface area of a sphere is 144 pie, then its radius is:
(A)6cm (B)8cm
(C)12cm (D)10cm
Answer: (A)6cm

144. Ques7.The circumference of a circle is 44cm. Then the area of the circle is:
(A)276cm square (B)44cm square
(C)176cm square (D)154cm square
(D)154cm square
Ques8.If sin 3A= Cos (A-26), where 3A is an acute angle find the value of
A.
(A)26 (B)27
(C)28 (D)29
Answer: (D)29

145. Ques9.Triangle ABC and triangle PQR are similar triangle such that angle
A=32 degree and angle R=65 degree then angle B is;
(A)83degree (B)32 degree
(C)65 degree (D)97 degree
Ques10.Which of the following are not the sides of a right triangle?
(A)9cm,15cm,12cm (B)2cm,1cm,10cm
(C)400cm,300cm,500cm (D)9cm,5cm,7cm
Answer;(D)9cm,5cm,7cm

146. Ques11.Solve for x and y : x / a + y / b =2 ax-by=a*a-b*b
(A) x=a ,y=b (B) x=b, y=a
(C) x=a-b ,y= a+ b (D) x=a + b, y=a-b
Answer; (A )x=a ,y=b
Ques12.Triangle ABC is similar to triangle PQR, in which
QP=3cm,QR=6cm,BC=8cm,AC=4 under root 3
(A)2+under root 3 (B)4+3under root 3
(C)4+under root 3 (D)3+4 under root 3
Answer; (B)4+3 under root 3.

147. Q-1 Find the area of square whose perimeter is 84.
ANS. Side=perimeter/4=84/4=21
Area=21 21=441
Q-2 Product of two numbers is 8192.If one number is
twice the other , find smallest number.
ANS. Let one number=x
Then, x 2x=8192
x x=4096
x=64
Q-3What will be next: 123,234,345,…..,……
ANS.456,567,678
Q-4 Average of four numbers is 30. If sum of 1
st
three
numbers is 85. Find fourth number.

148. ANS. 85+X/4=30
X=120-85
4
TH
no.=35
Q-5 The next line will be:
2550 53
2652 55
2754 57
ANS. 28 56 59
Q-6In a year the 1
st
April was Monday . What will
be the day on 18
th
April in the same year?
ANS. The day on 18
th
April will be Thursday.
Q-7 In figure 36490, digits 6 and 9 are replaced.
The difference between the new formed and
original number……..
ANS.39460-36490=2970

149. Q-8 What will be the next:
416 64
636 216
864 512
ANS. 10 100 1000
Q-9 Number of prime numbers between 10 and 20 is:
ANS. 4 prime numbers:
11, 13, 17, 19
Q-10 The length of a rectangle is 4m. The breadth is
half of it . What will be its perimeter?
ANS. Perimeter=2(L+B)
=2(4+2)
=2 6=12m

151. Wikipedia
Learnnext
Mypbworks.com
britannica.com
mathsisfun.com
math.com
libraryquest.org

152. REFRENCES
Applied Mathematics
Basic Mathematics
Calculus
Complex Analysis
Constants & Numerical
Sequences
Functional Analysis
R.D. Sharma
R.S. Aggarwal

153. Click here to download this powerpoint template : Brown Floral Background Free Powerpoint Template
For more : Templates For Powerpoint
Page 153
Thank You