TRIANGLES AND ITS PROPERTIES

PAGE# 83

TRIANGLES

AND ITS PROPERTIES

7

TRIANGLE

A triangle is a closed figure made by three line segments. It has three vertices, three sides and three angles.

A

B C

A triangle ABC is given having three vertices A, B, C and three sides AB, BC, CA and three angles are

ABC, BCA, CAB.

CLASSIFICATION OF TRIANGLES: 1. With respect to angles are:

(a) Acute-angled triangles (b) Obtuse-angled triangles (c) Right-angled traingles

(a) Acute-angled triangles

A

B C

A triangle in which measurement of each of the three angles are less than 90°.

(b) Obtuse-angled triangle :

A

B C

A triangle in which one angle is obtuse angle i.e. greater than 90° but less then 180°.

(c) Right-angled triangle:

A

B C

A triangle in which one angle is exactly 90°.

• The side opposite to the right angle is called the hypotenuse.

• The other two sides are called as the legs of the right-angled triangle (or base and perpendicular).

2. With respect to sides are:

(a) Scalene triangles (b) Isosceles triangles (c) Equilateral triangles

(a) Scalene Triangle: A triangle in which no two sides are equal in length is called a scalene triangle.

• Length of all sides and measurement of all angles is different. (b) Isosceles Triangle

An isosceles triangle is a triangle in which

A

B C

(i) Two sides have same length

(ii) The angles opposite to the equal sides are equal.

If in ABC, AB = AC then B = C this is an isoceles triangle. Converse is also true

If in a triangle two angles are equal then sides opposite to equal angles are equal in length. In ΔABC, if B = C then AB = AC.

(c) Equilateral Triangle In an equilateral triangle

(i) All sides are equal in length. (ii) Measurement of each angle is 60°.

A

B C

In ΔABC, if AB = BC = AC then ABC60

or if ABC60 then AB = BC = AC 3. Pythagoras Property of Right-Angled Triangle

In a right angled triangle, the square of the hypotenuse is equal to the sum of the squares of the legs.

A B C 2 2 2 _{BC AC} AB  

CONVERSE OF PYTHAGORAS THEOREM

In a triangle if the square of one side is equal to the sum of the squares of the remaining two sides then the angle opposite to the first side is a right angle.

It is clear

that:-AC2 _{> AB}2 _{and AC}2 _{> BC}2

 AC > AB and AC > BC.

Hence, we can say that in right triangle the hypotenuse is the largest side.

A

B C

4 cm 3 cm

PAGE# 85

A ABC is such that AB = 4 cm, AC = 3cm, BC = 5 cm and BC2 _{= AB}2 _{+ AC}2 _{ 5}2 _{= 4}2 _{+ 3}2_,

then measure of A should be 90°.

Note.: If the pythagoras property holds, the triangle must be right angled. Pythagorian Triplets

It is a set of three natural numbers a, b, c such that a2 _{+ b}2 _{= c}2_{. Such a set of three number it is called as}

Pythagorean Triplet. For example- (3, 4, 5), (5, 12, 13), (6, 8, 10) etc. PERIMETER OF A TRIANGLE

The sum of the lengths of the sides of a triangle is called its Perimeter.

A

B _a C

b c

In ΔABC, AC = b, BC = a, AB = c then perimeter is BC + CA + AB = a + b + c

TRIANGULAR INEQUALITY PROPERTY A

B C

The sum of the lengths of any two sides of a triangle is always greater than the third side. AB + BC > AC

AB + AC > BC AC + BC > AB

The difference between the length of any two sides of a triangle is smaller than the length of the third side. AB – BC < CA BC – CA < AB CA – AB < BC Example X Y Z 3 4 5 XY + YZ > XZ 3 + 4 > 5 similary YZ + XZ > XY 4 + 5 > 3 similary XY + XZ > YZ 3 + 5 > 4

MEDIANS OF A TRIANGLE AND CENTROID Medians

:-A

B _D C

Here BD = DC

AB, BC, CA are the sides of ΔABC and AD is a line segment which intersects side BC of ΔABC at

point D such that AD bisects the side BC. This line segment AD is called the median of ΔABC.

A median is a line segment which connects a vertex of a triangle to the mid point of the opposite side. Centroid

:-A

B _D C

F _G E

• Centroid is the point of intersection of all three medians of a triangle. • In the given figure G is the centroid of _ΔABC.

• A centroid of a triangle cuts the median in the ratio 2 : 1

 AG : GD = 2 : 1

ALTITUDES OF A TRIANGLE AND ORTHOCENTRE

Altitudes

:-A

B _D C

AD is a line segment which is perpendicular to the side BC of ΔABC.This line segment AD is called as

altitude of ΔABC.

Altitude is also called as height of a triangle. In the given figure AD is the height of ΔABC from vertex AA

to side BC. Orthocentre :-A B _D C E F H

•

Orthocentre is the meeting point of all three altitudes of a triangle.

•

In the given figure H is the orthocentre of ΔABC.

•

In obtuse-angled triangle the orthocentre lies outside the triangle.

•

In Acute-angled triangle the orthocentre lies inside the triangle.

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ANGLE BISECTORS OF A TRIANGLE AND IN-CENTRE Angle Bisectors

:-A

B _D C

Angle bisector of a triangle is a line segment which bisect the angle and whose end points lies on the vertex and its opposite side.

•

In the given figure, AD is the bisector of ΔABC.

So BADDAC Incentre A B _D C E F R Q I P

•

In-centre is the meeting point of all three angle bisectors of a triangle.

•

In the given figure, I is the Incentre of ABC

•

In-centre always lies inside the triangle.

•

The perpendicular distance from incentre to the side of triangle is always same.

 IP = IQ = IR

•

A circle can be drawn inside the triangle by assuming I as centre and IP as radius.

PERPENDICULAR BISECTOR AND CIRCUMCENTRE Perpendicular Bisector A B C O M N

It is a line segment which is perpendicular to the side of a triangle and passing through the mid point of the side.

Circumcentre

•

Circumcentre is the meeting point of all the perpendicular bisectors of a triangle

•

In the given figure ‘O’ is the circumcentre of ABC.

•

In an Acute-angle triangle. It lies inside the triangle.

•

In a right-angle triangle. It lies on the hypotenuse of the triangle. It is the mid point of hypotenuse.

•

In an obtuse-angle triangle. It lies outside and near the largest side of the triangle.

•

The distance between circumcentre and the vertices of a triangle is always same.

•

We can draw a circle passing through the vertices of triangle assuming ‘O’ as centre and OA as radius. ANGLE SUM PROPERTY OF A TRIANGLE:

The sum of the measures of three angles of a triangle is 180°. A

B C

1

2 ₃

 123180

Proof of Angle Sum Property of a Triangle

5 ₄ A P Q 1 3 2 C B

Given : A triangle ABC.

To prove : _{ABC180}

Construction : Draw a line segment PQ through A and parallel to BC. Proof: Mark the angles as indicated in the figure.

S.NO.

S

TATEMENT

R

EASON

1. 25 Alternate interior angles.

2. 3  4 Alternate interior angles.

3. 2354 Adding 1 and 2.

4. 123541 Adding 1 to both sides.

5. 123180 541180 by linear pair property.

Hence, ABC180

PAGE# 89

Exterior Angle Property of a Triangle

-1 2

A

B

C D

If we produce the side BC of a ABC then in the exterior of ABC an angle ACD is formed at point C. This ACD is called as an exterior angle of ABC formed at vertex C. The remaining two angles of

ABC

 are A and B. These two angles are called as interior opposite angles or the remote interior angles of ACD.

Exterior angle property states that an exterior angle of a triangle is equal to the sum of interior opposite angles.

 ACD1  2

Proof

:-Since, BC is a line which is produced to D

 BCAACD180 ...(i)

     

1 2 BCA 180 (Angle sum property of a triangle) ...(ii)

From (i) and (ii) we can say that

2 1    ACD Illustration 1

Is it possible to draw a triangle whose sides are 3 cm, 4 cm and 7 cm ? Solution

Sides of a triangle are 3 cm, 4cm, 7 cm. Here 3 + 4 = 7

and we know that sum of two sides of a triangles is always greater than the third side so it is not possible to draw a triangle whose sides are 3 cm, 4 cm and 7 cm.

Illustration 2

In each of the following there are three positive numbers. State if these numbers could possibly be the lengths of the sides of a triangle.

(i) 2, 3, 4 (ii) 2.5, 1.5, 4 Solution

(i) We have,

2 + 3 > 4, 2 + 4 > 3 and 3 + 4 > 2

That is, the sum of any two of the given numbers is greater than the third number. So, 2 cm, 3 cm and 4 cm can be the lengths of the sides of a triangle.

(ii) We have, 2.5 + 1.5 | 4.

Illustration 3

The length of two sides of a triangle are 12 cm and 15 cm. Between what two measure should the length of the third side fall ?

Solution

Let x cm be the length of the third side.

Then 12 + x > 15; 15 + x > 12 and 12 + 15 > x.  x > 15 – 12; x > 12 –15 and 27 > x.  x > 3 ; x > – 3 and 27 > x.

A number greater than 3 is obviously greater than – 3.  x > 3 and 27 > x.

Hence, x lies between 3 cm and 27 cm.

Property : Angles opposite to equal sides of a triangle are equal. Property : Sides opposite to equal angles of a triangle are equal.

Property : In a right triangle, if a, b are the lengths of the sides and c that of the hypotenuse, then c2 _{= a}2 _{+ b}2_.

(Hypotenuse)2 _{= (Base)}2 _{+ (Perpendicular)}2

Property : If the sides of a triangle are of lengths a, b and c such that c2 _{= a}2 _{+ b}2_{, then the triangle is right}

angled and the side of length c is the hypotenuse.

NOTE : Three positive numbers a, b, c in this order are said to form a pythagorean triplet, if c2 _{= a}2 _{+ b}2_{. Triplets (3, 4, 5) (5, 12, 13), (8, 15, 17), (7, 24, 25) and (12, 35, 37) are some pythagorean}

triplets. Illustration 1

The sides of certain triangles are given below. Determine which of them are right triangles : (i) a = 6 cm, b = 8 cm and c = 10 cm

(ii) a = 5 cm, b = 8 cm and c = 11 cm. Solution

(i) Here the larger side is c = 10 cm.

We have : a2 _{+ b}2 _{= 6}2 _{+ 8}2 _{= 36 + 64 = 100 = c}2_.

So, the triangle with the given sides is a right triangle. (ii) Here, the larger side is c = 11 cm

Clearly, a2 _{+ b}2 _{= 25 + 64 = 89  c}2_.

PAGE# 91

Illustration 2

A ladder 25 m long reaches a window of a building 20 m above the ground. Determine the distance of the foot of the ladder from the building.

Solution

Suppose that AB is the ladder, B is the window and CB is the building. Then, triangle ABC is a right triangle, with right angle at C.  AB2 _{= AC}2 _{+ BC}2  252 _{= AC}2 _{+ 20}2  AC2 _{= 625 – 400 = 225}  AC = 225 m = 15 m. Illustration 3

A ladder 17 m long reaches a window which is 8 m above the ground on one side of the street. Keeping its foot at the same point, the ladder is turned to the other side of the street to reach a window at a height of 15 m. Find the width of the street.

D

A _C B

E

8 cm 17 m 17 m 15 m

Solution

Let AB be the street and C be the foot of the ladder. Let D and E be the windows at the height of 8 m and 15m respectively from the ground.

Then, CD and CE are the two positions of the ladder. In right triangle, DAC, we have

AC2 _{+ AD}2 _{= CD}2

 AC2 _{= (CD}2 _{– AD}2₎

= {(17)2 _{– (8)}2_{} m}2 _{= 289 – 64 = 225 m}2

AC = 225 m = 15 m

In right triangle, CBE, we have CB2 _{+ BE}2 _{= CE}2

 CB2 _{= (CE}2 _{– BE}2_{) = {(17)}2 _{– (15)}2_{} m}2 _{= 64 m}2

CB = 64 m = 8 m

Illustration 4

A tree has broken at a height of 5 m from the ground and its top touches the ground at a distance of 12m from the base of the tree. Find the original height of tree.

12 m C A D B 5m Solution

Let AB be the tree and Let C be the point at which it broke. Then CB takes the position CD.

Original height of tree i.e., AB

i.e. AC + BC  AC + CD (BCCD)

i.e. ACD, using pythagoras theorm, we CD2 _{= AC}2 _{+ AD}2

CD2 _{= (5)}2 _{+ (12)}2 _{= 25 + 144 = 169}

CD2 _{= 13}2

CD = 13 m

So. height of tree = AC + BC = AC + CD = (5+13) m = 18 m

PAGE# 93 Q.1. Find x, when AB = BC. 140º A B C x

Q.2 Find the value of x and y in the given figure.

Q.3 Find the value of x and y in the given figure.

143º x y A B D C E 90º

Q.4 In given figure, AB divides DAC in the ratio of DAB : BAC = 1 : 3 and AB = DB. Find the value of x . B C D xº 108º A

Q.5 In ABC, 6 A = 4 B = 3 C, find the angles of ABC

Q.6 In figure, find value of x.

Q.7 The side BC of a ΔABC is produced to D. If the bisector of A meets BC at point L. Prove that ABC + ACD = 2ALC.

Q.8 ABCD is a quadrilateral. Is AB + BC + CD + DA > AC + BD ? Q.9 In the figure, find

D C B A E 20° 100° 25°

(i) ACD (ii) AED

Q.10 The sides BC, CA and AB of a ABC, are Produce in order, forming exterior angles,ACD, BAE and CBF. Show that ACDBAECBF360

E A B C D F 1 2 3

Q.11 _{In the following figure, final ADC.}

70°

B C D

PAGE# 95

Q.1 In the figure, ABCD is a rectangle, CEF is an equilateral triangle. Find x. [IMO-2013]

(A) 25° (B) 30° (C) 20° (D) 50°

Q.2 The given figure shows three identical squares. Find x. [IMO-2014]

(A) 30° (B) 27° (C) 36° (D) 16°

Q.3 In the given figure (not drawn to scale), ABCD is a square such that AE = DE. Find  EC.

[IMO-2015]

B _C

A E D

28°

(A) 28° (B) 56° (C) 62° (D) 24°

Q.4 In the given figure, if AC||OD and AO||EF. Then, measure of x and y respectively is : [IMO-2016]

O A C D F 75° y x E 20° B (A) 75°, 85° (B) 95°, 85° (C) 75°, 75° (D) 95°, 75°

Q.5 If AB  BC, BD  AC, CE bisects C and A = 30°, then CED = [IMO-2016] D A E B C (A) 30° (B) 60° (C) 45° (D) 65°

Q.6 In the given figure, find the value of x, [IMO-2016]

x 100° D 70° C 20° A B (A) 5° (B) 10° (C) 25° (C) 30°

PAGE# 97

SECTION - A FILL IN THE BLANKS

1. A triangle whose no two sides are equal, is called a _________ 2. Each angle of equilateral triangle is ______

3. An exterior angle is always _______ than either of the interior opposite angles . 4. Vertical angle of an isosceles triangle is 40 degree. Each of the base angle is ______. 5. A polygon with eight sides is known as ______ .

SECTION - B TRUE / FALSE

1. Traingle can be made from side 1cm , 2 cm, 3cm.

2. If the base angles of an isosceles triangle each measure 37 degrees, then the vertex angle has a measure of 106 degrees.

3. Sum of the measure of the three angles of an acute triangle is less than the sum of the measure of the three angles of an obtuse.

SECTION - C Q.1 For a triangle ABC which statement is always true :

(A) AC2 _{= AB}2 _{+ BC}2 _{(B) AC = AB + BC (C) AC > AB + BC (D) AC < AB + BC}

Q.2 Find the value of X in the figure given below :

(A) 118º (B) 18º (C) 108º (D) 128º

Q.3 If one angle of a triangle is equal to half the sum of the other two equal angles, the triangle is :

(A ) Equilateral (B) Isosceles (C) Right angled (D) Isosceles right angled

Q.4 In the diagram, what is the measure of ABC ?

60º 30º C A D B (A) 45º (B) 30º (C) 15º (D) 65º

Q.5 In ABC , AD bisects BAC and AD= DC. If ADB = 100º, then find ABD. A

B D C

100º

(A) 30º (B) 45º (C) 60º (D) 90º

Q.6 In right angled triangle ABC, EC is a bisector of the BCD and BD  AC. BAC = 30º, then CED is :

(A) 65º (B) 60º (C) 45º (D) 30º

Q.7 If one of the interior angles of a regular polygon is to be equal to       8 9

times of one of the interior angles of a regular hexagon, then the number of sides of the polygon is :

(A) 7 (B) 8 (C) 4 (D) 5

Q.8 If in ABC, AB = 5 cm, BC = 7 cm CA = 9 cm. Then which is correct

(A) _{ABC} (B) _{ABC} (C) _{BAC} (D) _{BCA} Q.9 In an isosceles triangle, measure of one angle is 110°, the other angles are ?

(A) 55°, 55° (B) 35°, 35° (C) 55°, 35° (D) None of these Q.10 If in ABC, AB = 5 cm, BC = 7 cm CA = 9 cm. Then which is correct

(A) ABC (B) ABC (C) BAC (D) BCA

Q.11 If the area of two triangles are equal and their sides are in the ratio 4 : 9 then the ratio of their corresponding altitudes are

PAGE# 99

Q.12 AB  BC, BD  AC and CE bisects C. If A = 30°. Then what is CEB.

30° A B C D E (A) 30° (B) 60° (C) 45° (D) 65°

Q.13 In the given figure A = 80°, B = 60°, C = 2x° and BDC = y°°, BD and CD bisects angles B and C respectively. The values of x and y respectively are

80° y° x° x° A B C D (A) 15°, 70° (B) 10°, 160° (C) 20°, 130° (D) 20°, 125° Q.14 AD, BE and CF are the medians of ABC. If GD = 1.5 cm. Then the length of AD is :

A B _D C E F G (A) 2.5 cm (B) 3.0 cm (C) 4.00 cm (D) 4.5 cm

Q.15 In a ABC,if the bisector of the angle BAC meets BC in D, then which one of the following is correct?

B D C

A

(A) _ABBD (B) BDA 1 BAC

2

  

SECTION - D MATCH THE COLUMNS

Q.1 Column – A Column–B

1. If two angles of triangle are 72° and 48°. a. 18° Then 3rd _{angle is}

2. one acute angle of right triangle is 72°. b. 60° Find other acute

3. point of concurrence of 3 altitudes c. Centroid

4. point of concurrence of 3 medians d. Orthocenter

5. If 3 angles of a triangle are x, x + 12, x – 12. Then x is

Q.2 Column – A Column–B

1. Polygon with eight sides a. Convex

2. Each angle of equilateral triangle b. 180°

3. Sum of all angles of triangle c. octagon

4. If all interior angles of polygon are d. 60° less than 180 degree

PAGE# 101

ANSWERS

[Subjective Questions] 1. 40º 2. xº = 45º, yº = 26º 3. x = 53º, y = 127º 4. 90º 5. A = 400_{, B = 60}0_{, C = 80}0 6. 80º 8. Yes 9. (i) 125° (ii) 145° 11. 27.5°

[Previous Years Questions]

1. A 2. B 3. B 4. A 5. B 6. B

SECTION - A

1. scalene 2. 60° 3. greater 4. 70°

5. octagon

SECTION - B

1. FALSE 2. TRUE 3. FALSE

SECTION - C 1. D 2. C 3. A 4. C 5. A 6. B 7. B 8. C 9. B 10. C 11. A 12. B 13. C 14. D 15. B SECTION - D 1. 1-(b); 2-(a); 3-(d); 4-(c); 5-(b) 2. 1-(c); 2-(d); 3-(b); 4-(a)