Download Triangles - Study Point

6
Triangles
CHAPTER
We are Starting from a Point but want to Make it a Circle of Infinite Radius
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
B  900 .
(ii)
Acute angle triangle a triangle in which all angles
are less than 900 in measures is called acute angled
triangle.
011-26925013/14
NTSE, NSO
+91-9811134008 Diploma, XI Entrance
+91-9582231489
85
Triangles
(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)
(f)
Angle bisectot: the angle bisector of an angle of a
triangle is a line that divided the angle in two equal
part
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‟.
011-26925013/14
NTSE, NSO
+91-9811134008 Diploma, XI Entrance
+91-9582231489
86
Triangles
(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)
(h)
Perpendicular bisector: the perpendicular
bisector of a triangle is perpendicular drawn from
the opposite vertex and divide the opposite side in
two equal parts.
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 , B  75 , C  32 find A .
3.
In the figure, show that
0
0
A  B  C  D  E  F  3600
011-26925013/14
NTSE, NSO
+91-9811134008 Diploma, XI Entrance
+91-9582231489
87
Triangles
4.
In the figure, prove that a  b  c  360
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 128 0 and 220 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 =
0
900 +
8.
1
A .
2
In ABC , B > C , if AM is the bisector of BAC and AN  BC, prove that MAN =
B  C ).
1
(
2
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 380, 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 .
Given A
To prove
Proof.
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.
ABC whose side BC has been produced to D forming exterior angle ACD .
A  B  ACD
In
ABC
….(1)
….(2) (straight angle)
A  B  ACB  1800
ACB  ACD  1800
From (1) and (2), we get
ACB  ACD  A  B  ACB
ACD  A  B
011-26925013/14
NTSE, NSO
+91-9811134008 Diploma, XI Entrance
+91-9582231489
88
Triangles
A  B  ACD
IX ACADEMIC QUESTIONS
Subjective
Assengment-2
1.
In the figure, find BED .
2.
In figure, prove that x  A  B  C .
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.
011-26925013/14
NTSE, NSO
+91-9811134008 Diploma, XI Entrance
+91-9582231489
89
Triangles
7.
If ABD = 1250 and ACE = 1300, 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. BAC  FDE
(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 ABC and DEF
If
AB = DE, A  D ,
AC = DF
Then ABC  DEF .
011-26925013/14
NTSE, NSO
+91-9811134008 Diploma, XI Entrance
+91-9582231489
90
Triangles
Rule 2 (SSS). When three sides are given
In ABC and DEF
If
AB = DE, BC = EF, CA = FD
Then ABC  DEF .
Rule 3 (ASA). When two angles and the included side is given
In ABC and DEF
If
B  E , BC = EF, C  F
Then ABC  DEF .
Rule 4 (R.H.S). When Right Angle – Hypotenuse – Side are given
In ABC and DEF
If
B  E = 900, BC = EF, CA = FD
Then ABC  DEF .
Congruence Relations of Triangles
(i)
Reflexive ABC  ABC (congruence relation is reflexive) Every triangle is congruent to itself.
(ii) Commutative If ABC  DEF , then DEF  ABC (congruence relation is commutative)
(iii) Transitive If ABC  DEF , and DEF  PAR then ABC  PAR (congruence relation
is transitive)
Note :
Since the congruence relation is reflexive, commutative and transitive, it is an equivalence
relation.
011-26925013/14
NTSE, NSO
+91-9811134008 Diploma, XI Entrance
+91-9582231489
91
Triangles
IX ACADEMIC QUESTIONS
Subjective
Asseingment-3
1.
Prove that ABC is isosceles if altitude AD bisects BC.
2.
Prove that ABC
perpendicular to BC.
3.
In the fig. it is given that AB = CF, EF = BD and AFE =
DBC . Prove that AFE  CBD .
4.
ABCD is a quadrilateral in which AD = BC and DAB  CBA . Prove that :
is isosceles if median AD is
(i) ABD  BAC
(ii) BD = AC
(iii) ABD  BAC
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.
011-26925013/14
NTSE, NSO
+91-9811134008 Diploma, XI Entrance
+91-9582231489
92
Triangles
7.
In fig. It is given that BC = CE and 1  2 . Prove that
GCB  DCE .
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
BAD  CBE
ECA  DCB
Prove that (i) DAC  EBC (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, TPS  QPR . 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 DAQ  CBP .
011-26925013/14
NTSE, NSO
+91-9811134008 Diploma, XI Entrance
+91-9582231489
93
Triangles
14.
In the fig. AB = AC, ABD  ACD Prove that BD = CD.
15.
In fig. QPR  PQR 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 EBC  FCB .
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.
011-26925013/14
NTSE, NSO
+91-9811134008 Diploma, XI Entrance
+91-9582231489
94
Triangles
(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 ABD  ACD .
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.
011-26925013/14
NTSE, NSO
+91-9811134008 Diploma, XI Entrance
+91-9582231489
95
Triangles
IX ACADEMIC QUESTIONS
Subjective
Assignment-4
1.
2.
3.
Prove that the difference of any two sides of a triangle is less than the third.
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.
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
PSR  PSQ .
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. x  y , show that M  N .
011-26925013/14
NTSE, NSO
+91-9811134008 Diploma, XI Entrance
+91-9582231489
96
Triangles
XI SCIENCE & DIP. ENTRANCE
Subjective
Assignment (MCQ) – 5
1.
In the figure AB||CD, ABE  128 0 , BED  20 0 , what is the value of EDC ?
(a) 128 o
(b) 148 o
(c) 108 o
2.
3.
(d) 130 o
In the given figure AB||CD, EFC  30 0 and ECF  100 0 , then BAF is equal to
(a) 70 0
(b) 80 0
(c) 100 0
(d) 130 0
In  ABC, when BC is produced on both ways, the exterior angles are 102 0 and 134 0 , what is
the value of A
(a) 36 0
(b) 56 0
(c) 106 0
(d) 1120
4.
In the figure calculate the value of „y‟
(a) 45 0
(b) 50 0
0
(c) 60
(d) 90 0
5.
In the figure the value of
(a) 100 0
(c) 115 0
6.
7.
is
(b) 109 0
(d) 120 0
The angles of a triangle are 2x  10 , x  20 and
triangle it is?
(a) Equilateral Triangle
(b) Right Angled Triangle
(c) Acute Angle Triangle
(d) Obtuse Angle Triangle
0
0
x  100 ,
which type of
In the trapezium ABCD, EF||AD, what is the value of ACD ?
(a) 70 0
(c) 50 0
8.
x0
if x  5 0
(b) 60 0
(d) 40 0
In the figure CE is perpendicular to AB. ACE  20 0 and ABD  50 0 , what is the measure of
BDA ?
(a) 50 0
(b) 60 0
(c) 70 0
(d) 90 0
011-26925013/14
NTSE, NSO
+91-9811134008 Diploma, XI Entrance
+91-9582231489
97
Triangles
9.
In the figure what is the relation of
x in term of a, b and c ?
(a) x  a  b  c
(b) x  a  b  c  90
(c) x  180   a  b  c 
(d) x  180  a  b  c 
10. PN
 QR and PM is bisector of P in  PQR, then ( Q  R ) is equal to
(a) MPN
1
(c) MPN
2
(b) 2MPN
1
(d) MPN
3
11. The bisector of exterior angles B and C meet at „O‟, what is BOC
(a) 120 0
(b) 80 0
(c) 45 0
(d) 40 0
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 BPC  BQC is equal to
(a) 90 0
(b) 90  1 / 2A
(c) 90  1 / 2A
(d) 180 0
15. In the figure what is the value of a  b  c ?
(a) 90 0
(b) 180 0
(c) 270 0
(d) 360 0
16. In the figure PM is bisector of P and PN is perpendicular on QR
then the value of MPN is
(a) 30 0
(b) 40 0
60 0
(d) 100
(c)
0
17. BM and CM are interior bisectors of B and
B and C respectively. Which is correct?
C
while BN and CN are exterior bisectors of
(a) BMC  110 0
(b) BNC  70 0
(c) BMC  BNC  180 0
011-26925013/14
NTSE, NSO
+91-9811134008 Diploma, XI Entrance
+91-9582231489
98
Triangles
(d) All are correct
18. In a ABC, AB = 5cm, AC = 5cm and A = 50 o, then B =
(a) 35 o
(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 = 48 o, 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
011-26925013/14
NTSE, NSO
+91-9811134008 Diploma, XI Entrance
+91-9582231489
99
Triangles
(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 = 37o, y = 70o
(c) x + y = 117o
(d) x – y = 100o
30. In the given figure BD  AC, the measure of ABC is
(a) 60 o
(b) 30 o
(c) 45 o
011-26925013/14
NTSE, NSO
+91-9811134008 Diploma, XI Entrance
+91-9582231489
(d) 90 o
100
Triangles
ANSWER
Assignment – 1
2. A = 71 o
5. 30 o, 50 o , 100 o
9. (a) 180 o
(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
6. 52 o, 53 o , 75 o
Assignment – 2
1., BED = 82 o 3. x = 60 o, y = 55 o
5. x = 115 o , y = 20 o
6. x = 40 o, y = 20 o
7. 75 o
Assignment – 3
20. 90
21. (i) equal
(ii) equal
(vi) Equal to
(iii) 60
(iv) BC
(v) AC
(ii) True
(iii) True
(iv) True
(v) True
(vii) True
(viii) True
(vii) EFD
22. (i) False
(vi) 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
011-26925013/14
NTSE, NSO
+91-9811134008 Diploma, XI Entrance
+91-9582231489
101
Triangles