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CHAPTER

6We are Starting from a Point but want to Make it a Circle of Infinite Radius

Triangles

A plane figure bounded by three line segments is called a triangle.

We denote a triangle by the symbol . In fig. ABC has

(i) three vertices namely, A,B and C

(ii) three sides namely, AB, BC, CA

(iii) three angles namely, A , B and C .

Types of Triangles on the basis of sides

(i) Equilateral triangle. A triangle whose all the three

sides are equal is called equilateral.

In the figure ABC is an equilateral triangle in

which AB = BC = CA

(ii) Isosceles triangle. A triangle having two sides

equal is called an isosceles triangle.

In the figure, ABC is an isosceles triangle in which

AB = AC.

(iii) Scalene triangle. A triangle whose sides are of

different lengths. In the figure ABC is a triangle in

which AB BC CA.

Types of Triangles on the Basis of Angles

(i) Obtuse-angled triangle. A triangle in which one

angle is an obtuse angle, is called an obtuse angled

triangle. In figure, ABC is a triangle in which 090B .

(ii) Acute angle triangle a triangle in which all angles

are less than 900 in measures is called acute angled

triangle.

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(iii) A right angled triangle : A triangle in which one

angle is of exact 900 is called right angle triangle.

Some Other Important Terms of Triangles

(a) Median. A median of a triangle is the line segment

joining the mid-point of side with the opposite

vertex.

(b) Centroid. The point of intersection of all the three

medians of a triangle is called its centroid.

Characteristics of Centroid

(i) Centroid is the point at which the medians of triangle

meet.

(ii) The medians of a triangle are concurrent.

(iii) The centroid divides the medians in the ratio 2 : 1.

(iv) The median of an equilateral triangle are equal.

(v) The medians of an equilateral triangle coincide with

the “altitudes”.

(c) Altitudes. The altitude of a triangle is the

perpendicular drawn from a vertex to the opposite

side.

(d) Orthocentre. The point of intersection of all the

three altitudes of a triangle is called its orthocenter.

Characteristics of Orthocentre

(i) Orthocentre is the point at which the altitudes of a

triangle meet.

(ii) The altitudes of a triangle are concurrent.

(iii) Orthocentre of an acute triangle lies in the interior of

the triangle.

(iv) Orthocentre of an obtuse triangle lies in the exterior of

the triangle.

Orthocentre of a right triangle lies on the

vertex of the right angle.

(e) Angle bisectot: the angle bisector of an angle of a

triangle is a line that divided the angle in two equal

part

(f) Incentre of a triangle. The point of intersection of

the bisectors the internal angles of a triangle in

called its incentre.

Characteristics of Incentre

(i) The point at which the three angle bisectors of a

triangle intersect is called the „incentre‟.

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(ii) The triangle may be acute, obtuse, or right, the angle

bisectors of a triangle must meet at a point lying inside

the triangle.

(iii) The incentre of a triangle lies in the interior of the

triangle.

(iv) The bisectors of the angles of a triangle are

concurrent.

(v) From the incentre we can draw a on opposite sides.

(vi) We can call this perpendicular as “inradius”

(g) Perpendicular bisector: the perpendicular

bisector of a triangle is perpendicular drawn from

the opposite vertex and divide the opposite side in

two equal parts.

(h) Circumcentre of a triangle. The point of

intersection of the perpendicular bisectors of the

sides of a triangle is called its circumcentre.

Characteristics of Circumcentre

(i) The point at which the perpendicular bisectors of the

sides of a triangle meet is called the cicumkcentre of

the triangle.

(ii) The right bisectors of the sides of a triangle are

concurrent.

(iii) With circumcentre as centre, we can drawn a circle

passing through the vertices of a triangle.

(iv) The distance from centre to the vertices is called the

„circumradius‟.

The circle thus drawn with circumentre as centre

and circumradius as radius is called

„circumcircle‟.

(i) Perimeter. The sum of lengths of the sides of a figure is its perimeter. Perimeter of ABC = AB

+ BC + CA.

Asseingment-1

1. Prove that the sum of the angles of a triangle is 1800.

2. In ABC , 075B ,

032C find A .

3. In the figure, show that 0360 FEDCBA

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4. In the figure, prove that 0360 cba

5. The angles of a triangle are in the ratio 3 : 5 : 10. Find the measures of each angle of the triangle.

6. The sum and difference of two angles of a triangle are 1280 and 22

0 respectively. Find all the angles

of the triangle.

7. If the bisector of the angle B and C of a ABC meet at a point O, then prove that BOC =

900 +

2

1 A .

8. In ABC , B > C , if AM is the bisector of BAC and AN BC, prove that MAN = 2

1 (

CB ).

9. Fill in the blanks

(a) The sum of three angles of a triangle is ……..

(b) If two angles of a triangle are 510 and 38

0, the third angle is equal to ……….

(c) If the angles of a triangle in the ratio 2 : 2 : 5, then the angles are……..

(d) The angles of a triangle are 3x – 5, 2x + 55 and 5x – 50 degrees then x is equal to…………

(e) A triangle cannot have more than……………..right angles.

(f) A triangle cannot have more than…………obtuse angle.

10. Which of the following statements are true or false?

(a) An exterior angle of a triangle is less than either of its interior opposite angles.

(b) Sum of the three angles of a triangle is 1800.

(c) A triangle can have two right angles.

(d) A triangle can have two acute angles.

(e) A triangle can have two obtuse angles.

(f) An exterior angle of a triangle is equal to the sum of the two interior opposite angles.

Exterior angle of a Triangle:

Definition. If the side BC of a triangle ABC is

produced to ray BD, then ACD is called an

exterior angle of triangle ABC at C, and is denoted

by exterior ACD . A and B are called remote

interior angles or interior opposite angles.

Note : At each vertex there are two exterior angles

Theorem . If a side of a triangle is produced, then the exterior angle so formed is equal to

the sum of the two interior opposite angles.

Given A ABC whose side BC has been produced to D forming exterior angle ACD .

To prove ACDBA

Proof. In ABC

0180 ACBBA ….(1)

0180 ACDACB ….(2) (straight angle)

From (1) and (2), we get

ACBBAACDACB

BAACD

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ACDBA

IX ACADEMIC QUESTIONS Subjective

Assengment-2

1. In the figure, find BED .

2. In figure, prove that CBAx .

3. In the figure, find x and y, if AB || DF and AD ||

FG.

4. In the figure, prove that DE || BF.

5. In the figure, AB || DG, AC || DE, EDH = 250

and BAC = 200, Find x and y.

6. In the figure, CD AB, ABE = 1300 and

BAC = 700. Find x and y.

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7. If ABD = 1250 and ACE = 130

0, then BAC

= …….

Congruent Figures

The geometric figures are said to be congruent if they are exactly of the same shape and size.

(a) Congruent segments. Two segments are congruent if they are of the same length and conversely.

Hence in Fig. AB CD

A B

C D

(b) Congruent angles. Two angles are congruent if they have equal measures and conversely.

Hence in fig. FDEBAC

(c) Congruent circles. Two circles are congruent if they have equal radii and conversely.

Hence in fig.

If r1 = r2

Then c1 circle c2 circle

Rules for Congruent triangles

Rules 1. (SAS) When two sides and the included angle are given

In DEFandABC

If AB = DE, DA , AC = DF

Then DEFABC .

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Rule 2 (SSS). When three sides are given

In DEFandABC

If AB = DE, BC = EF, CA = FD

Then DEFABC .

Rule 3 (ASA). When two angles and the included side is given

In DEFandABC

If EB , BC = EF, FC

Then DEFABC .

Rule 4 (R.H.S). When Right Angle – Hypotenuse – Side are given

In DEFandABC

If EB = 900, BC = EF, CA = FD

Then DEFABC .

Congruence Relations of Triangles

(i) Reflexive ABCABC (congruence relation is reflexive) Every triangle is congruent to itself.

(ii) Commutative If DEFABC , then ABCDEF (congruence relation is commutative)

(iii) Transitive If DEFABC , and PARDEF then PARABC (congruence relation

is transitive)

Note : Since the congruence relation is reflexive, commutative and transitive, it is an equivalence

relation.

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IX ACADEMIC QUESTIONS Subjective

Asseingment-3

1. Prove that ABC is isosceles if altitude AD bisects BC.

2. Prove that ABC is isosceles if median AD is

perpendicular to BC.

3. In the fig. it is given that AB = CF, EF = BD and AFE =

DBC . Prove that CBDAFE .

4. ABCD is a quadrilateral in which AD = BC and CBADAB . Prove that :

(i) BACABD

(ii) BD = AC

(iii) BACABD

5. In the fig. AC = AE and AB = AD and BAD = EAC .

Prove that BC = DE.

6. In fig. l || m and M is the mid point of the line segment AB.

Prove that M is also the mid-point of nay line segment CD

having its end points on l and m respectively.

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7. In fig. It is given that BC = CE and 21 . Prove that

DCEGCB .

8. In fig. AD and BC are perpendicular to the line segment AB

and AD = BC. Prove that O is the mid point of line segment

AB and DC.

9. In the figure C is the mid point of AB

CBEBAD

DCBECA

Prove that (i) EBCDAC (ii) DA = EB

10. In the fig. BM and DN are both perpendiculars to the

segments AC and BM = DN. Prove that AC bisects BD.

11. In fig., PS = PR, QPRTPS . Prove that PT = PQ.

12. In fig. AD = AE and D and E are points on BC

such that BD = EC. Prove that AB = AC.

13. In the fig. AD CD and BC CD. If AQ = BP

and DP = CQ. Prove that CBPDAQ .

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14. In the fig. AB = AC, ACDABD Prove that BD = CD.

15. In fig. PQRQPR and M and N are respectively on

sides QR and PR of PQR such that QM = PN. Prove that

OP = OQ, where O is the point of intersection of PM and

QN.

16. In the fig. AB = AC. BE and CF are respectively the

bisectors of B and C . Prove that FCBEBC .

17. AD and BE are respectively altitude of ABC such that AE = BD. Prove that Ad = BE.

18. AD, BE and CF, the altitudes of ABC are equal. Prove that ABC is an equilateral triangle.

19. AD is the bisector of A of a triangle ABC, P is any point on AD. Prove that the perpendicular

drawn from P on AB and AC are equal.

20. ABCD is a parallelogram, if the two diagonals are equal, find the measure of ABC .

21. Fill in the blanks in the following so that each of the following statements is true.

(i) Sides opposite to equal angles of a triangle are………….

(ii) Angle opposite to equal sides of a triangle are…………

(iii) In an equilateral triangle all angles are…………..

(iv) In a ABC if A = C , then AB = ……….

(v) If altitudes CE and BF of a triangle ABC are equal, then AB = ……….

(vi) In an isosceles triangle ABC with AB = AC, if BD and CE are its altitudes, then BD is

…………. CE.

(vii) In right triangles ABC and DEF, if hypotenuse AB = EF and side AC = DE, then

ABC ……………

22. Which of the following statements are true and which are false.

(i) Sides opposite to equal angles of a triangle are unequal.

(ii) Angles opposite to equal sides of a triangle are equal.

(iii) The measure of each angle of an equilateral triangle is 600.

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(iv) If the altitude from one vertex of a triangle bisects the opposite side, then the triangle is

isosceles.

(v) The bisectors of two equal angles of a triangle are equal.

(vi) If the bisector of the vertical angle of a triangle bisects the base, then the triangle may be

isosceles.

(vii) If any two sides of a right triangle are respectively equal to two sides of other right triangle,

then the two triangles are congruent.

(viii) Two right triangles are congruent if hypotenuse and a side of one triangle are respectively

equal to the hypotenuse and a side of the other triangle.

23. ABC and DBC are two isosceles triangles on the same base BC. Show that ACDABD .

24. If ABC is an isosceles triangle with AB = AC. Prove that the perpendicular from the vertices B

and C to their opposite sides are equal.

25. If the altitudes from two vertices of a triangle to the opposite sides are equal. Prove that the triangle

is isosceles.

26. In a right angled triangle, one acute angle is double the other. Prove that the hypotenuse is double

the smallest side.

Inequalities in a Triangle

(i) If two sides of a triangle are unequal, the longer side has greater angle opposite to it.

(ii) If two angles of a triangle are unequal, the greater angle has the longer side opposite to it.

(iii) The sum of any two sides of a triangle is greater than the third side.

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IX ACADEMIC QUESTIONS Subjective

Assignment-4

1. Prove that the difference of any two sides of a triangle is less than the third.

2. Show that of all the line segments that can be drawn to a given line from a given point not lying on

it, the perpendicular line segment is the shortest.

3. In a right angled triangle, prove that the hypotenuse is the longest side.

4. In the figure, AD is the bisector of A , show that AB > BD.

5. In figure PR > PQ and PS bisects QPR . Prove that

PSQPSR .

6. In the fig. PQ > PR. QS and RS are the bisector of Q and

R respectively. Prove that SQ > SR.

7. In the fig. yx , show that NM .

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XI SCIENCE & DIP. ENTRANCE Subjective

Assignment (MCQ) – 5

1. In the figure AB||CD, 0128ABE ,

020BED , what is the value of EDC ?

(a) 128o

(b) 148o

(c) 108o

(d) 130o

2. In the given figure AB||CD, 030EFC and

0100ECF , then BAF is equal to

(a) 070 (b)

080

(c) 0100 (d)

0130

3. In ABC, when BC is produced on both ways, the exterior angles are 0102 and

0134 , what is

the value of A

(a) 036 (b)

056

(c) 0106 (d)

0112

4. In the figure calculate the value of „y‟ if 05x

(a) 450 (b) 50

0

(c) 600

(d) 900

5. In the figure the value of 0x is

(a) 1000

(b) 1090

(c) 1150

(d) 1200

6. The angles of a triangle are 010x2 , 020x and 010x , which type of

triangle it is?

(a) Equilateral Triangle (b) Right Angled Triangle

(c) Acute Angle Triangle (d) Obtuse Angle Triangle

7. In the trapezium ABCD, EF||AD, what is the value of ACD ?

(a) 070 (b)

060

(c) 050 (d)

040

8. In the figure CE is perpendicular to AB. 020ACE and

050ABD , what is the measure of

BDA ?

(a) 050 (b)

060

(c) 070 (d)

090

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9. In the figure what is the relation of x in term of ,a b and c ?

(a) cbax (b) 90cbax

(c) x 180 a b c

(d) cba180x

10. PN QR and PM is bisector of P in PQR, then ( RQ ) is equal to

(a) MPN (b) MPN2

(c) MPN2

1

(d) MPN

3

1

11. The bisector of exterior angles B and C meet at „O‟, what is BOC

(a) 0120 (b)

080

(c) 045 (d)

040

12. The bisectors of exterior angles of ABC intersect at „O‟ and form a BOC which is always

(a) Acute Angle

(b) Right Angle

(c) Obtuse Angle

(d) None of these

13. The bisectors of interior angles of a triangle forms an angle which is always

(a) Acute Angle

(b) Right Angle

(c) Obtuse Angle

(d) All of these

14. In a triangle ABC, the internal bisectors of angles B and C meet at „P‟ and the external

bisectors of the angles B and C meet at Q, then the BQCBPC is equal to

(a) 090 (b) A2/190

(c) A2/190 (d) 0180

15. In the figure what is the value of cba ?

(a) 090 (b)

0180

(c) 0270 (d)

0360

16. In the figure PM is bisector of P and PN is perpendicular on QR

then the value of MPN is

(a) 030 (b)

040

(c) 060 (d)

0100

17. BM and CM are interior bisectors of B and C while BN and CN are exterior bisectors of

B and C respectively. Which is correct?

(a) 0110BMC

(b) 070BNC

(c) 0180BNCBMC

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(d) All are correct

18. In a ABC, AB = 5cm, AC = 5cm and A = 50o, then B =

(a) 35o (b) 65

o (c) 80

o (d) 40

o

19. If two sides of a triangle are unequal then opposite angle of larger side is

(a) greater (b) less (c) equal (d) half

20. The sum of attitudes of a triangle is_______ than the perimeter of the triangle

(a) greater (b) less (c) half (d) less

21. In the given figure, PQ = QR, QPR = 48o, SRP = 18

o, then PQR =

(a) 48

o (b) 84

o (c) 30

o (d) 36

o

22. In the given figure, PQR is an equilateral triangle and QRST is a square. Then PSR =

(a) 30

o (b) 15

o (c) 90

o (d) 60

o

23. Can we draw a triangle ABC with AB = 3cm, BC = 3.5cm and CA = 6.5cm?

(a) Yes (b) No (c) Can‟t be determined (d) None of these

24. Which of the following is not a criterion for congruence of triangles?

(a) SSA (B) SAS (c) ASA (d) SSS

25. In the given figure, AB BE and EF BE. Also BC = DE and AB = EF. Then

(a) BD = FEC (b) ABD = EFC (c) ABD = CMD (d) ABD = CEF

26. In quadrilateral ABCD, BM and DN are drawn perpendicular to AC such that BM = DN. If BR

= 8cm, then BD is

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(a) 4cm (b) 2cm (c) 12cm (d) 16cm

27. In the given figure PQ > PR, QS and RS are the bisectors of Q and R respectively.

(a) SQ = SR (b) SQ > SR (c) SQ < SR (d) None of these

28. In the figure, PS is the median, bisecting angle P, then QPS is_______

(a) 110

o (b) 70

o (c) 45

o (d) 55

o

29. In the given figure x and y are

(a) x = 70

o, y = 37

o (b) x = 37

o, y = 70

o (c) x + y = 117

o (d) x – y = 100

o

30. In the given figure BD AC, the measure of ABC is

(a) 60

o (b) 30

o (c) 45

o (d) 90

o

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ANSWER

Assignment – 1

2. A = 71o 5. 30

o, 50

o, 100

o 6. 52

o, 53

o, 75

o

9. (a) 180o (b) 91

o (c) 40

o, 40

o, 100

o (d) 18

o (e) one (f) one

10. (a) False (b) True (c) False (d) True (e) False (f) True

Assignment – 2

1., BED = 82o 3. x = 60

o, y = 55

o 5. x = 115

o, y = 20

o 6. x = 40

o, y = 20

o

7. 75o

Assignment – 3

20. 90

21. (i) equal (ii) equal (iii) 60 (iv) BC (v) AC

(vi) Equal to (vii) EFD

22. (i) False (ii) True (iii) True (iv) True (v) True

(vi) True (vii) True (viii) True

Assignment – 5

1.c 2.d 3.b 4.b 5.d 6.b 7.a 8.b 9.c 10.d 11.c 12.a

13.c 14.d 15.d 16.a 17.d 18.b 19.a 20.b 21.b 22.b 23.b 24.a

25.a 26.d 27.b 28.c 29.b 30.d