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Guided effort leads to a bright futureUTUT

IX-XII

CBSE ICSE

State

Entrance

X - ICSE - MATHSVOLUME 2

Table of Contents

Volume 2 of 2 Universal Tutorials – X ICSE – Mathematics

Table of Contents

CHAPTER 15: SIMILARITY (WITH APPLICATIONS TO MAPS AND MODELS) .... 1

Chapter Map ...................................................................................................................................... 1 Introduction: ....................................................................................................................................... 1

Similarity of figures: ...................................................................................................................... 1 Similarities of triangles: ................................................................................................................ 1 Three similarity postulates for triangles (Axioms of similarity of triangles): ................................. 1

Solved Example 15.1: .............................................................................................................. 2 Unsolved Exercise 15.1: .......................................................................................................... 3 Unsolved Exercise 15.2: .......................................................................................................... 5

Relation between the areas of two similar triangles:.................................................................... 6 Solved Examples 15.3: ............................................................................................................ 8 Unsolved Exercise 15.3: .......................................................................................................... 9 Previous Board Questions: .................................................................................................... 11 Miscellaneous Exercise: ........................................................................................................ 12 Answers to Unsolved Exercise .............................................................................................. 14

CHAPTER 16: LOCI (LOCUS AND ITS CONSTRUCTIONS) ................................ 16

Chapter Map: ................................................................................................................................... 16 Introduction: ..................................................................................................................................... 16

Locus: ......................................................................................................................................... 16 Solved Example 16.1: ............................................................................................................ 18 Unsolved Exercise 16.1: ........................................................................................................ 18

Important Points to Remember: ...................................................................................................... 20 Solved Examples 16.2: .......................................................................................................... 21 Unsolved Exercise 16.2: ........................................................................................................ 22 Previous Board Question: ...................................................................................................... 24 Answers to Unsolved Exercise: ............................................................................................. 25

CHAPTER 17: CIRCLES ......................................................................................... 27

Chapter Map .................................................................................................................................... 27 Introduction: ..................................................................................................................................... 27

Chord and its properties: ............................................................................................................ 28 Arc and its Chord Properties: ..................................................................................................... 28 Segment and relation between arcs and segments: .................................................................. 28

Theorem Based on .......................................................................................................................... 29 Angle properties ......................................................................................................................... 29 Cyclic Properties: ....................................................................................................................... 30

Solved Example 17.1: ............................................................................................................ 31 Unsolved Exercise 17.1: ........................................................................................................ 31

Arc and chord properties: ........................................................................................................... 38 Solved Example 17.2: ............................................................................................................ 39 Unsolved Exercise 17.2: ........................................................................................................ 40 Previous Board Question: ...................................................................................................... 41 Miscellaneous Exercise: ........................................................................................................ 43 Answer to Unsolved Exercise: ............................................................................................... 45

CHAPTER 18: TANGENTS AND INTERSECTING CHORDS ................................ 47

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Universal Tutorials – X ICSE – Mathematics Volume 2 of 2


Solved Example 18.1: ............................................................................................................ 50 Unsolved Exercise 18.1: ........................................................................................................ 50 Solved Example 18.2: ............................................................................................................ 53 Unsolved Exercise 18.2: ........................................................................................................ 54 Previous Board Questions: .................................................................................................... 55 Miscellaneous Exercise: ........................................................................................................ 57 Answers to Unsolved Exercise: ............................................................................................. 60

CHAPTER 19: CONSTRUCTIONS ......................................................................... 61


Tangent: ..................................................................................................................................... 61 Construction of tangents to a given circle: ................................................................................. 61 Construction of circumscribed and inscribed circles of a triangle: ............................................. 62

Unsolved Exercise: ................................................................................................................ 63 Previous Board Questions: .................................................................................................... 64 Miscellaneous Exercise: ........................................................................................................ 65 Answers to Unsolved Exercise: ............................................................................................. 66

CHAPTER 20: CYLINDER CONE AND SPHERE (SURFACE AREA AND VOLUME) .................................................................................................................. 67


Cylinder Review: ........................................................................................................................ 67 Solved Example 20.1: ............................................................................................................ 68 Unsolved Exercise 20.1: ........................................................................................................ 68

Cone: .......................................................................................................................................... 69 Solved Examples 20.2: .......................................................................................................... 69 Unsolved Exercise 20.2: ........................................................................................................ 69

Sphere: ....................................................................................................................................... 70 Spherical Shell: .......................................................................................................................... 70 Hemi–sphere: ............................................................................................................................. 70

Solved Examples 20.3: .......................................................................................................... 71 Unsolved Exercise 20.3: ........................................................................................................ 71 Solved Example 20.4: ............................................................................................................ 72

Conversion of solids: .................................................................................................................. 72 Unsolved Exercise 20.4: ........................................................................................................ 73 Solved Example 20.5: ............................................................................................................ 74 Unsolved Exercise 20.5: ........................................................................................................ 75

Cross–Sectional Problems: ........................................................................................................ 75 Solved Example 20.6: ............................................................................................................ 75 Unsolved Exercise 20.6: ........................................................................................................ 76 Previous Board Question: ...................................................................................................... 77 Miscellaneous Exercise: ........................................................................................................ 77 Answers to Unsolved Exercise: ............................................................................................. 80

CHAPTER 21: TRIGONOMETRICAL IDENTITIES ................................................. 82


Table of Contents

Volume 2 of 2 Universal Tutorials – X ICSE – Mathematics

Trigonometry: ............................................................................................................................. 82 Relations between different trigonometrical ratios: .................................................................... 82

Solved Example 21.1: ............................................................................................................ 83 Unsolved Exercise 21.1: ........................................................................................................ 84 Solved Example 21.2: ............................................................................................................ 85 Unsolved Exercise 21.2: ........................................................................................................ 85 Solved Examples 21.3: .......................................................................................................... 87 Unsolved Exercise 21.3: ........................................................................................................ 87

Using the Trigonometrical Tables: ............................................................................................. 89 Solved Example 21.4: ............................................................................................................ 89 Unsolved Exercise 21.4: ........................................................................................................ 89 Previous Board Question: ...................................................................................................... 90 Miscellaneous Exercise: ........................................................................................................ 91 Answers to Unsolved Exercise: ............................................................................................. 93

CHAPTER 22: HEIGHTS AND DISTANCES .......................................................... 94


Practical Use of Trigonometry: ................................................................................................... 94 Angles of elevation and depression: .......................................................................................... 94

Solved Example 22.1: ............................................................................................................ 94 Unsolved Exercise 22.1: ........................................................................................................ 95 Solved Example 22.2: ............................................................................................................ 96 Unsolved Exercise 22.2: ........................................................................................................ 96 Previous Board Question: ...................................................................................................... 97 Miscellaneous Exercise: ........................................................................................................ 98 Answers to Unsolved Exercise: ........................................................................................... 101

CHAPTER 23: GRAPHICAL REPRESENTATION (HISTOGRAMS AND OGIVES) ................................................................................................................................. 102

Chapter Map: ................................................................................................................................. 102 Introduction: ................................................................................................................................... 102

Frequency Distribution: ............................................................................................................ 102 Graphical Representation (Histograms and Ogives): .............................................................. 103 Histogram: ................................................................................................................................ 103 Histogram for continuous grouped data: .................................................................................. 103 Cumulative Frequency Curve or an Ogive: .............................................................................. 103

Solved Example 23: ............................................................................................................. 104 Histogram for discontinuous grouped data: ............................................................................. 104

Unsolved Exercise 23: ......................................................................................................... 105 Previous Board Questions: .................................................................................................. 107 Answers to Unsolved Exercise: ........................................................................................... 111

CHAPTER 24: MEASURES OF CENTRAL TENDENCY (MEAN, MEDIAN, QUARTILES AND MODE) ....................................................................................... 112

Chapter Map: ................................................................................................................................. 112 Introduction: ................................................................................................................................... 112

Arithmetic Mean: ...................................................................................................................... 112 Arithmetic Mean of Tabulated Data:......................................................................................... 113 To find Mean of Grouped Data (Both continuous and discontinuous): .................................... 113

Solved Examples 24.1: ........................................................................................................ 115

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Universal Tutorials – X ICSE – Mathematics Volume 2 of 2

Unsolved Exercise 24.1: ...................................................................................................... 115 Median: ..................................................................................................................................... 117 Median for Raw and Arrayed Data: (tabulated) ....................................................................... 117 Quartiles: .................................................................................................................................. 117

Solved Example 24.2: .......................................................................................................... 118 Unsolved Exercise 24.2: ...................................................................................................... 120

Mode: ....................................................................................................................................... 121 Solved Exercise 24.3: .......................................................................................................... 121 Unsolved Exercise 24.3: ...................................................................................................... 122 Previous Board Questions: .................................................................................................. 122 Miscellaneous Exercise: ...................................................................................................... 125 Answer to Unsolved Exercise: ............................................................................................. 127

CHAPTER 25: PROBABILITY .............................................................................. 129

Introduction: .............................................................................................................................. 129 Some Basic Terms and Concepts ................................................................................................. 129

Experiment: .............................................................................................................................. 129 Random Experiment: ............................................................................................................... 129 Sample Space: ......................................................................................................................... 129 Equally Likely Outcomes: ......................................................................................................... 130

Measurement of Probability:.......................................................................................................... 130 Empirical (or experimental) Probability: ................................................................................... 131 Classical (or theoretical) Probability: ........................................................................................ 131 An Event: .................................................................................................................................. 131

Solved Examples 25.1: ........................................................................................................ 132 Unsolved Exercise 25.1: ...................................................................................................... 133

Points to be Remember: .......................................................................................................... 135 Solved Examples 25.2: ........................................................................................................ 136 Unsolved Exercise 25.2: ...................................................................................................... 137 Previous Year Board Questions: ......................................................................................... 138 Miscellaneous Exercise: ...................................................................................................... 139 Answers for the Unsolved Exercise: .................................................................................... 142

BOARD QUESTION PAPER ................................................................................. 144

Latest ICSE Board Question Paper [2017–18] .................................................................... 144 UT Sample Paper Based on (2017–18) ICSE Curriculum with Model Answer ................... 149

Chapter 15: Similarity (With Applications to Map and Models) 1

Volume 2 of 2 Universal Tutorials – X ICSE – Mathematics 1

Chapter 15: Similarity (With Applications to Maps and Models)

Chapter Map

Introduction:

Similarity of figures:

Any two figures which have exactly the same shape, but not necessarily the same size, are called similar figures.

Similarities of triangles:

When two triangles are similar; their corresponding angles are equal and corresponding sides are proportional.

For example:

If ABC is similar to DEF,

i.e. ABC DEF;

A = D, B = E, C = F,

and DE

AB =

EF

BC =

DF

AC.

Note: The sign ‘’ is read as, ‘is similar to’.

Three similarity postulates for triangles (Axioms of similarity of triangles):

1) If two triangles have a pair of corresponding angles equal and the sides including them proportional; then the triangles are similar by SAS postulate.

For example:

In ABC and DEF,

If B = E and DE

AB =

EF

BC,

Then ABC DEF.

A

B C

D

E F

A

B C

D

E F

Introduction Similarity of

triangles

Relation between the areas of two similar triangles (Theorem

and applications

Similarity (With Applications to Maps and Models)

Applications to maps

and models

Application of basic proportionality

theorem

3 Similarity

Postulates

2

2 Universal Tutorials – X ICSE – Mathematics Volume 2 of 2

2) If two triangles have two pairs of corresponding angles equal; the triangles are similar by AA or AAA postulate.

For example :

In ABC and DEF.

If A = D and B = E,

Then ABC DEF.

3) If two triangles have their three pairs of corresponding sides proportional, the triangles are similar by SSS postulate.

Eg. In ABC and DEF.

If DE

AB =

EF

BC =

DF

AC, then ABC DEF

Note:

1) In ABC, with A = 90 and AD BC.

We get three pairs of similar triangles.

a) ABD CBA [By AA]

BC

AB =

AB

BD AB

2 = BD BC

b) ABC DAC [By AA]

DC

AC =

AC

BC AC

2 = DC BC

c) BAD ACD [By AA]

DC

AD =

AD

BD AD

2 = BD DC

SOLVED EXAMPLE 15.1:

1) In the fig, given below, the medians BD and CE of a triangle ABC meet at G. Prove that:

i) EGD ~ CGB and ii) BG = 2GD and CG = 2 EG. Sol: The two medians BD and CE meet at G. Also E and D are the mid points of AB and AC respectively.

EB

AE =

DC

AD = 1,

AB

AE =

AC

AD =

2

1

(i)

i) In AED and ABC,

A

B C

D

E F

A

B D C

A

B D

C

A

B

A

B C

D C

A

A

B D C

A

B

90 – x

90 – x

x

B

A

C

G

D E



AB

AE =

AC

AD and EAD = BAC

AED ABC Proved

ED || BC (converse of B.P.T.)

Also, BC

ED =

AC

AD =

2

1 (from (i) above) (ii)

ii) In EGD and CGB

EGD = CGB (vertically opp. angles)

EDG = GBC (alt. angles)

GED = GCB (alt. angles)

Hence EGD CGB (AAA postulates)

CG

EG =

BG

GD =

BC

ED=

2

1 (from (ii) above)

CG = 2 EG, BG = 2 GD (Proved).

UNSOLVED EXERCISE 15.1:

CW Exercise

1) State, true or false:

i) Two similar polygons are necessarily congruent.

ii) Two congruent polygons are necessarily similar.

iii) All equiangular triangles are similar

iv) All isosceles triangles are similar

v) Two isosceles-right triangles are similar

vi) Two isosceles triangles are similar, if an angle of one is congruent to the corresponding angle of the other.

vii) The diagonals of a trapezium, divide each other into proportional segments.

2) Given: GHE = DFE = 90,

DH = 8, DF = 12, DG = 3x – 1 and DE = 4x + 2.

Find: the lengths of segments DG and DE

3) Two right triangles ABC and DBC are drawn on the same

side of hypotenuse BC. If AC and DB intersect at P.

Prove that: AP PC = BP PD

4) Given: RS and PT are altitudes of PQR. Prove that:

i) PQT ~ RQS ii) PQ QS = RQ QT.

5) Given: FB = FD, AE FD and FC AD.

Prove: ED

BC

AD

FB

6) In PQR, Q = 90 and QM is perpendicular to PR. Prove that:

i) PQ2 = PM PR ii) QR

2 = PR MR iii) PQ

2 + QR

2 = PR

2.

7) In ABC, B = 90 and BD AC.

i) If CD = 10 cm and BD = 8cm; find AD. ii) If AC = 18cm and AD = 6 cm; find BD.

iii) If AC = 9cm and AB = 7cm; find AD.

8) In quadrilateral ABCD, diagonals AC and BD intersect at point E such that: AE : EC = BE : ED. Show that ABCD is a trapezium.

H D

F

E

G

D

C

P

B

A

B C A D

E

F

4


9) In triangle ABC, AD is perpendicular to side BC and AD2 = BD DC. Show that angle BAC = 90.

10) In the given figure, QR is parallel to AB and DR is parallel to QB.

Prove that: PQ2 = PD PA.

11) In the figure given below P is a point on AB

such that AP : PB = 4 : 3, PQ is parallel to AC.

i) Calculate the ratio PQ : AC, giving reason for your answer.

ii) In triangle ARC, ARC = 90 and in triangle PQS, PSQ = 90.

Given QS = 6cm, calculate the length of AR.

12) In the right angled triangle QPR, PM is an altitude

Given that QR = 8cm and MQ = 3.5cm.

Calculate the value of PR.

13) In the figure, given below, straight line AB and CD intersect at

P. and AC BD. Prove that:

i) APC and BPD are similar.

ii) If BD = 2.4 cm, AC = 3.6 cm, PD = 4.0 cm and PB = 3.2 cm, find the lengths of PA and PC.

14) In the given figure, DE BC, AE = 15 cm, EC = 9 cm, NC = 6 cm and BN = 24 cm

i) Write all possible pairs of similar triangles.

ii) Find lengths of ME and DM.

15) In the given figure, AB DC, BO = 6 cm

and DQ = 8 cm, find BP DO.

HW Exercise:

1) In triangle ABC, DE is parallel to BC; where D and E are the points on AB and AC respectively.

Prove ADE ~ ABC. Also, find the length of DE, if AD = 12 cm, BD = 24 cm and BC = 8cm.

2) D is a point on the side BC of triangle ABC such that angle ADC is equal to angle BAC. Prove

that: CA2 = CB CD.

3) E and F are the points in sides DC and AB respectively of parallelogram ABCD. If diagonal AC

and segment EF intersect at G. Prove that: AG EG = FG CG.

4) Given: ABCD is a rhombus,

DPR and CBR are straight lines.

Prove that: DP CR = DC PR.

B A

D

Q R

P

B A

C

Q

R

P

S

B A

C

F

D

E

P

R

R Q

P

M 3.5 cm

8 cm



5) In ABC, B = 2C and the bisector of angle B meets CA at point D. Prove that:

i) ABC and ADB are similar, ii) DB : AD = BC : AB.

6) In ABC, right-angled at C, CD AB. Prove: CD2 = AD DB

7) In the figure, PQRS is a parallelogram with

PQ = 16cm and QR = 10cm. L is a point on PR

such that RL : LP = 2:3. QL produced meets RS

at M and PS produced at N.

Find the lengths of PN and RM

8) Given: AB || DE and BC || EF.

Prove that:

i) FG

CF

DG

AD

ii) DFG ~ ACG.

9) In the given figure, AB || EF || DC;

AB = 67.5 cm, DC = 40.5 cm and AE = 52.5 cm.

i) Name the three pairs of similar triangles

ii) Find the length of EC and EF.

10) Through the mid-point M of the side CD of a parallelogram ABCD, the line BM is drawn intersecting diagonal AC in L and AD produced in E. Prove that: EL = 2BL.

Basic Theorem of Proportionality:

A line drawn parallel to any side of a triangle, divides the other two sides proportionally. (Basic Proportionality Theorem or Thales Theorem).

In the given figure, DE || BC BD

AD =

CE

AE

Conversely: If a line divides two sides of a triangle proportionally, the line is parallel to the third side

i.e. if BD

AD =

CE

AE DE || BC

b) In the same figure, given above

ADE ABC [By AAA postulate]

AB

AD =

AC

AE =

BC

DE.


Class Work: 1) In the following figure, point D divides AB in the ratio 3:5. Find:

i) EC

AE ii)

AB

AD iii)

AC

AE

Also if:

iv) DE = 2.4 cm, find the length of BC.

v) BC = 4.8 cm, find the length of DE.

2) A line PQ is drawn parallel to the side BC of ABC which cuts side AB at P and side AC at Q. if AB = 9.0 cm, CA = 6.0 cm and AQ = 4.2 cm, find the length of AP.

R

M

L

Q

P N S

B

A C

F D

E

G

B

A

C F

D E

A

B C

D E

6


3) In the given figure, ABC ~ ADE. If AE : EC = 4:7 and DE = 6.6 cm, find C. if „x‟ be the length of the perpendicular from A to DE, find the length of perpendicular from A to BC in terms of „x‟.

4) In the figure, given below, AB, CD and EF are parallel lines. Given AB = 7.5 cm, DC = y cm, EF

= 4.5 cm, BC = x cm and CE = 3 cm, calculate the values of x and y. 5) In the following figure, M is mid–point of BC of a parallelogram ABCD.

DM intersects the diagonal AC at P and AB produced at E. prove that: PE = 2PD.

Home Work:

1) In the given figure, PQ AB; CQ = 4.8 cm, QB = 3.6 cm and AB = 6.3 cm, find:

i) PA

CP ii) PQ

iii) If AP = x, then the value of AC in term of x.

2) In ABC, D and E are the points on sides AB and AC respectively.

Find whether DE BC, if: i) AB = 9 cm, AD = 4 cm, AE = 6 cm and EC = 7.5 cm. ii) AB = 6.3 cm, EC = 11.0 cm, AD = 0.8 cm and AE = 1.6 cm.

3) A line segment DE is drawn parallel to base BC of ABC in which cuts AB at point D and AC at point E. if AB = 5 BD and EC = 3.2 cm, find the length of AE.

4) In the figure, given below, PQR is a right angle triangle right angled at Q. XY is parallel to QR, PQ = 6 cm, PY = 4 cm and PX : XQ = 1:2. Calculate the lengths of PR and QR.

5) The given figure shows a parallelogram ABCD. E is a point AD and

CE produced meets BA produced at point F. if AE = 4 cm, AF = 8 cm and AB = 12 cm, find the perimeter of the parallelogram ABCD.

Relation between the areas of two similar triangles:

Theorem 1:

The areas of two similar triangles are proportional to the squares on their corresponding sides.

Given: ABC DEF

Such that BAC = EDF

B = E and C = F.

To Prove: DEF Area

ABC Area

=

2

2

DE

AB =

2

2

EF

BC =

2

2

DF

AC

Construction: Draw AM BC and DN EF.

A

B C

D

E F M N



Proof:

1) Area of ABC = 2

1BC AM Area of =

2

1 base altitude

Area of DEF = 2

1 = EF DN Area of =

2

1 base altitude

DEFArea

ABCArea

=

DNEF

AMBC

2121

= EF

BC

DN

AM –– (i)

2) In ABM and DEN:

i) B =E [Given]

ii) AMB =DNE [Each angle being 90]

ABM DEN [By AA postulate]

DN

AM =

DE

AB –– (ii) [Corr. sides of similar are in proportion]

3) Since, ABC DEF [Given]

DE

AB =

EF

BC =

DF

AC –– (iii) [Corr. sides of similar are in proportion]

DN

AM =

EF

BC [From ii and iii]

Substituting DN

AM =

EF

BC in equation (i) we get

DEFArea

ABCArea =

EF

BC

EF

BC =

2

2

EF

BC –– (iv)

Now combining iii and iv, we get, DEFArea

ABCArea

=

2

2

DE

AB =

2

2

EF

BC =

2

2

DF

AC

Remember:

1) Median divides the triangle into two triangles of equal area.

In the given figure, AD is median

Area of ABD = Area of ACD = 2

1 Area of ABC

2) If many triangles have the common vertex and their bases are along the same straight line, the ratio between their areas is equal to the ratio between the lengths of their bases.

In the given figure, all the triangles have the common vertex at point A and bases of all the triangles are along the same straight line BC.

ADCArea

ABEArea

=

DC

BE,

ABCArea

ABDArea

=

BC

BD and so on

Applications of maps and models:

Maps and Models: Students use map of India, map of Asia etc in Geography. Consider the map of India in which the positions of major cities of India are shown. Measure the distance between any two cities marked in the map and compare it with the actual distance between those two cities to get the ratio between the two distances.

In the same way, choose two more cities marked in the map and compare their distance (on the map) with the actual distance between them to get the ratio between the two distance; we find that the ratio of distances in both the cases is the same.

A

B C D E

A

B C D

8


Infact, this ratio is already marked on every map and is known as scale–factor which is denoted by letter, „k‟.

For example:

If the scale of a map is 1: 20,000.this implies a distance of one cm on the map is equal to an actual distance of 20,000 cm (0.2 km) on the ground. And also scale factor

k = 000,20

1.

The same principle is applicable to models. In case of models,

object of Height

model the of Height =

object the of Length

model the of Length =

object the of Width

model the of Width = scale factor (k)

Important:

If the scale factor is k, then:

i) Each side of the resulting figure (the image) is k times the corresponding side of the given figure (the object or the pre–image).

ii) The area of the resulting figure is k2 times the area of the given figure.

iii) In case of solids, the volume of the resulting figure is k3 times the volume of the given figure.

iv) If the scale factor k > 1 the transformation is an enlargement and the image is larger than the object.

If the scale factor k < 1 the transformation is a reduction and the image is smaller than the object.

If the scale factor k = 1 the transformation is an identity transformation and the image is congruent to the object.

SOLVED EXAMPLES 15.3:

1) A model of a ship is made to a scale of 1 : 200.

i) The length of the model is 4m; calculate the length of the ship.

ii) The area of the deck of the ship is 160000 m2, find the area of the deck of the model.

iii) The volume of the model is 200 litres; calculate the volume of the ship in m3.

Sol: i) ship of Length

model of Length = scale factor

ship of Length

4 =

200

1

Length of ship = 800 m

ii) ship of deck of Area

model of deck of Area = (Scale factor)

2

160000

model of deck of Area =

2

200

1

Area of deck of model = 4 m2

iii) 3

3

m in ship of Volume

m in model of Volume = (Scale factor)

3

3

3

m in ship of Volume

m2.0 =

3

200

1

Volume of ship = (200)3 0.2 = 1600000 m

3

2) Two equilateral triangles have their areas in the ratio of 4:25. If each side of the smaller triangle is 6 cm, find the length of each side of the bigger triangle.

Sol: ABC ~ DEF (AA similarity) (both triangles are equilateral)

Let DEF

ABC

=

25

4

2

2

DE

AB=

25

4

DE

AB =

5

2

AB = 6 cm DE

6 =

5

2 DE = 15cm

A

B C

6 cm

60 cm

60 cm

D

E F

60 60



3) In ABC, B = 90, AB = 12 cm and AC = 15 cm. D and E are points on AB and AC

respectively such that AED = 90 and DE = 3 cm.

a) Find the area of ADE

b) Find )ABC(ar

)BCEDalqadrilater(ar

Sol: In ABC and AED, A is common and ABC = AED = 90

ABC ~ AED, by criterion of similarity.

AE

AB =

DE

BC =

AD

AC = k (say)

Now, BC = 22 ABAC = 222 )1215( cm = 2)1215)(1215( cm = 2327 cm

= 243 cm = 32 cm = 9 cm

Now, ar(ABC) = 2

1BC AB =

2

1 9 12 cm

2 = 54 m

2

Also, DE

BC = k

3

9 = k k = 3

a) Now, )(

)(

ADEar

ABCar

=

2

2

AE

AB = k

2

)(

54 2

ADEar

cm

= 3

2

ar(ADE) = 9

54 cm

2 ar(ADE) = 6 cm

2

b) ar(quadrilateral BCED) = ar(ABC) – ar(ADE) = 54 cm2 – 6 cm

2 = 48 cm

2

)(

)(

ABCar

BCEDralquadrilatear

=

2

2

54

48

cm

cm =

9

8


CW Exercise

1) a) The ratio between the corresponding sides of two similar triangles is 2 is to 5. Find the ratio between the areas of these triangles.

b) Areas of two similar triangles are 98 sq. cm and 128 sq. cm. Find the ratio between the lengths of their corresponding sides.

2) The perimeters of two similar triangles are 30 cm and 24 cm. If one side of first triangle is 12 cm, determine the corresponding side of the second triangle.

3) ABC is a triangle, PQ is a line segment intersecting AB in P and AC in Q such that PQ || BC and divides triangle ABC into two parts equal in area. Find the value of ratio BP : AB.

4) The given diagram shows two isosceles triangles which are similar

also. In the given diagram, PQ and BC are not parallel;

PC = 4, AQ = 3. QB = 12, BC = 15 and AP = PQ. Calculate:

a) the length of AP.

b) the ratio of the areas of triangle APQ and triangle ABC.

5) In the given figure, BC is parallel to DE. Area of ABC = 25 cm2,

Area of trapezium BCED = 24 cm2 and DE = 14 cm.

Calculate the length of BC.

Also, find the area of triangle BCD.

A

C

D

E

B

15cm

12cm

3cm

A

E

B C

D

B

C

Q

P A

10


6) On a map, drawn to a scale of 1:250000, a triangular plot PQR of land has the following

measurements: PQ = 3 cm, QR = 4 cm and angle PQR = 90. Calculate:

i) The actual lengths of QR and PR in kilometre.

ii) The actual area of the plot in sq. km.

7) In the given figure, B = E, ACD = BCE, AB = 10.4 cm

and DE = 7.8 cm. Find the ratio between areas of the ABC and DEC.

8) An aeroplane is 30 m long and its model is 15 cm long. If the total outer surface area of the model is 150 cm

2, find the cost of painting the outer surface of the aeroplane at the rate of

Rs.120 per sq. m Given that 50 sq. m of the surface of the aeroplane is left for windows.

9) A triangle ABC with AB = 3 cm, BC = 6 cm and AC = 4 cm is enlarged to DEF such that the

longest side of DEF = 9 cm. Find the scale factor and hence, the lengths of the other sides of

DEF.

10) Two isosceles triangles have equal vertical angles. Show that the triangles are similar. If the ratio between the areas of these two triangles is 16: 25, find the ratio between their corresponding altitudes.

HW Exercise

1) A line PQ is drawn parallel to the base BC of ABC which meets sides AB and AC at points P

and Q respectively. If AP = 31 PB; find the value of:

a) APQ of Area

ABC of Area

b)

PBCQ trap. of Area

APQ of Area

2) In the given figure AX : XB = 3 : 5, Find

i) the length of BC, if length of XY is 18 cm.

ii) ratio between the areas of trapezium XBCY and ABC.

3) In the given triangle PQR, LM is parallel to QR and PM : MR = 3 : 4.

Calculate the value of ratio:

i) PQ

PL and then

QR

LM ii)

MNR of Area

LMN of Area

iii)

LQN of Area

LQM of Area

4) In the figure, given below, ABCD is a parallelogram P is a point on BC such that BP : PC = 1 : 2, DP produced meets AB produced at Q. Given the area of triangle CPQ = 20 cm

2.

Calculate: i) Area of triangle CDP.

ii) Area of parallelogram ABCD.

5) The given figure shows a trapezium in which AB is parallel to DC and diagonals AC and BD intersect at point P. If AP : CP = 3 : 5.

Find:

i) APB : CPB ii) DPC : APB

iii) ADP : APB iv) APB : ADB

6 Triangle ABC is an isosceles triangle in which AB = AC = 13 cm

and BC = 10 cm. AD is perpendicular to BC.

If CE = 8 cm and EF AB, find:

i) FEBofarea

ADCofarea

ii)

ABCofarea

FEBofarea

A

C

D

E

B

A

E C D B

F

A

C

X Y

B



PREVIOUS BOARD QUESTIONS:

1) In the given figure, DE || BC.

i) Prove that ADE and ABC are similar. ii) Given that AD = ½ BD, calculate DE, if BC = 4.5 cm. [2004]

iii) also find )(

)(

ABCar

ADEar

and

)(

)(

BCEDtrapeziumar

ADEar

2) In the given figure, AB and DE are perpendicular to BC. If AB = 9 cm, DE = 3 cm and AC = 24 cm, calculate AD. [2005]

3) In the figure given, PB and QA are perpendiculars to the line segment AB. If PO = 6 cm, QO = 9 cm

and the area of POB = 120 cm2,

find the area of QOA. [2006]

4) In the given figure, ABC is a triangle. DE is parallel to BC and 2

3

DB

AD [2007]

i) Determine the ratios BC

DE

AB

AD,

ii) Prove that DEF is similar to CBF. Hence, find FB

EF.

iii) What is the ratio of the areas of DEF and BFC?

5) In ABC, AP : PB = 2:3 PO is parallel to BC and is extended to Q so that CQ is parallel to BA. Find: [2008]

i) area APO: area ABC

ii) area APO: area CQO

6) In the figure AB = 7 cm

and BC = 9 cm.

i) Prove ACD ~ DCB ii) Find the length of CD [2009]

7) The model of a building is constructed with scale factor 1 : 30. i) If the height of the model is 80 cm, find the actual height of the building in metres.

ii) If the actual volume of a tank at the top of the building is 27 m3, find the volume of the tank on

the top of the model. [2009]

8) In the given fig, ABC and CEF are two triangles where BA is parallel to CE and AF : AC = 5 : 8

i) Prove that ADF CEF.

ii) Find AD, if CE = 6 cm

iii) If DF is parallel to BC find area of ADF: area of ABC. [2009]

B

C

P A

M

A

B C

D E

F

A B

D

C

A

F E

C B

D

A

B

Q O

C

P

C

A

B

D

E P

B A

Q

O

12


9) In the given figure ABC is a triangle with EDB = ACB.

Prove that ABC ~ EBD [2010]

If BE = 6 cm, EC = 4 cm, BD = 5 cm and area of BED = 9cm2.

Calculate the

i) length of AB ii) area of ABC 10) In the adjoining figure ABC is a right

Angled triangle with BAC = 90 [2011]

i) Prove ADB ~ CDA. ii) If BD = 18 cm, CD = 8 cm find AD.

iii) Find the ratio of the area of ADB is to area of CDA.

11) In the given figure, ABC and AMP are right angled at B and M respectively.

Given, AC = 10 cm, AP = 15 cm and PM = 12 cm.

i) Prove ABC ~ AMP

ii) Find AB and BC. [2012]

12) In the given figure, AB and DE are perpendicular to BC.

i) Prove that ABC DEC.

ii) If AB = 6 cm, DE = 4 cm and AC = 15 cm, calculate CD.

iii) Find the ratio of the area of ABC: area of DEC. [2013]

13) In ABC, ABC=DAC, AB = 8cm, AC = 4cm, AD = 5cm,

i) prove that ACD is similar to BCA.

ii) find BC and CD.

iii) find area of ACD : area of ABC. [2014]

14) ABC is a right angled triangle with ABC = 90. D is any

point on AB and DE is perpendicular to AC. Prove that:

i) ADE ACB.

ii) If AC= 13cm, BC= 5 cm and AE = 4 cm. Find DE and AD.

iii) Find area of ADE: area of quadrilateral BCED. [2015]

15) A model of a ship is made to a scale 1:300. [2016]

i) The length of the model of the ship is 2m. Calculate the length of the ship.

ii) The area of the deck of the ship is 180,000 m2. Calculate the area of the deck of the model.

iii) The volume of the model is 6.5 m3. Calculate the volume of the ship.

MISCELLANEOUS EXERCISE:

1) In the given figure, lines l, m and n are such that l || m || n. Prove that:

QR

PQ

BC

AB

2) In the following figure, AD and CE are medians

of ABC. DF is drawn parallel to CE.

Prove that:

i) EF = FB

ii) AG : GD = 2 : 1

B

C

P A

M

A B

C

D

18cm

8cm

A

E B C

D



3) If two similar triangles are equal in area. Prove that the triangles are congruent.

4) The ratio between the areas of two similar triangles is 16:25. Find the ratio between their:

i) perimeters ii) altitudes iii) medians

5) In the following figure, AB, CD and EF are parallel lines.

AB = 6 cm, CD = y cm, EF = 10 cm, AC = 4 cm and CF = x cm.

Calculate x and y.

6) The dimensions of the model of a multistoreyed building are 1 m by 60 cm by 1.20 m. If the scale factor is 1 : 50, find the actual dimensions of the building. Also, find:

i) The floor area of a room of the building, if the floor area of the corresponding room in the model is 50 sq. cm.

ii) The space (volume) inside a room of the model, if the space inside, the corresponding room of the building is 90 m

3.

7) In the given triangle P, Q and R are the mid-points of sides AB, BC and AC respectively. Prove that triangle PQR is similar to triangle ABC.

8) In the given figure, triangle ABC is similar to triangle PQR. AM and PN are altitudes whereas AX and PY are medians.

Prove that: PY

AX

PN

AM

9) The ratio between the altitudes of two similar triangles is 3 : 5; write the ratio between their:

i) medians, ii) perimeters iii) areas

10) The following figure shows a triangle PQR in which XY is parallel to QR.

If PX : XQ = 1 : 3 and QR = 9 cm, find the length of XY.

Further, if area of PXY = x cm2; find the area of:

i) triangle PQR ii) trapezium XQRY

11) On a map, drawn to a scale of 1 : 20000, a rectangular plot of land ABCD has AB = 24 cm and BC = 32 cm. Calculate: i) the diagonal distance of the plot in kilometer (ii) the area of the plot in sq. km.

12) In a triangle PQR, L and M are two points on the base QR, such that

LPQ = QRP and RPM = RQP. Prove that:

i) PQL ~ RPM

ii) QL RM = PL PM

iii) PQ2 = QR QL

13) In ABC, ACB = 90 and CD AB. Prove that: AD

BD

AC

BC

2

2

14) A triangle ABC with AB = 3 cm, BC = 6 cm and AC = 4 cm is enlarged to DEF such that the

longest side of DEF = 9cm. Find the scale factor and hence, the lengths of the other sides of

DEF.

15) Two isosceles triangle have equal vertical angles. Show that the triangles are similar.

If the ratio between the areas of these two triangles is 16:25, find the ratio between their corresponding altitudes.

P

Q R L M

14


16) In ABC, AP :PB = 2:3. PO is parallel to BC and is extended

to Q so that CQ is parallel to BA.

Find i) Area APO : area ABC

ii) area APO : area CQO

17) The perimeters of two similar triangles are 36 cm and 24 cm respectively. If the shortest side of the second triangles is 5 cm long, find the length of the shortest side of the first triangle.

18) The following figure shows a triangle ABC in which AD and BE are perpendiculars to BC and AC respectively. Show that

i) ADC ~ BEC

ii) CA CE = CB CD

iii) ABC ~ DEC

iv) CD AB = CA DE

ANSWERS TO UNSOLVED EXERCISE

CW Exercise 15.1:

1) i) False ii) True iii) True iv) False v) True vi) True vii) True

2) 20 and 30 7) i) 6.4cm ii) 8.48cm iii) 59

4cm.

11) i) 3:7 ii) 14cm. 12) 6 cm

13) PA = 4.8 cm, PC = 6 cm (ii) ME = 3.7 cm and DM = 15 cm

14) ADM ~ ABN, AME ~ ANC, ADE ~ ABC

15) 48 cm2

HW Exercise 15.1:

1) 22/3cm

7) PN = 15cm and RM = 102/3cm

9) i) ABE and DEC; ABC and EFC; BEF and BDC (ii) 31.5 cm and 255/16 cm.

CW Exercise 15.3:

1) i) 4:25 ii) 7:8 2) 9.6cm

3) 2:12 4) a) 5cm b) 1:9

5) BC =10cm; 10cm2 6) i) QR = 10km. and PR =12.5km ii) 37.5km

2

7) 16:9 8) Rs.66,000

9) Scale factor = 1.5, DE = 4.5 cm; DF = 6 cm 10) 4:5

HW Exercise 15.3

1) i) 16:1 ii) 1:15 2) i) 48cm ii) 55:64

3) i) 3:7; 3:7 ii) 3:7 iii) 10:7 4) i) 40cm2 ii) 120cm

2

5) i) 3:5 ii) 25:9 iii) 5:3 iv) 3:8 6) i) 169 : 324; ii) 162 : 169

B

A

P O

Q

C

C

A B

E D



Previous Board Question:

1) DE = 1.5 cm, 9

1and

8

1 2) AD = 16 cm

3) 270 cm2 4) (i)

AB

AD =

5

3=

BC

DE (ii)

FB

EF =

5

3(iii)

25

9

5) (i) 25

4 (ii)

9

4 6) CD = 12 cm

7) (i) 24 m (ii) 1000 cm3 8) (ii) 10 cm (iii) 25:64

9) (i) 12 cm (ii) 36 cm2 10) (ii) 12 cm (iii) 9:4

11) (ii) BC = 8 cm; AB = 6 cm 12) (ii) CD = 10 cm (iii) 9:4

13) (ii) BC = 6.4, CD = 2.5 (iii) 1:4 14)DE = 1.67, AD= 4.33 (iii) 3.34 cm2, 26.67 cm

2

15) (i) 600 m (ii) 2m2 (iii) 17,55,00,000m

3

Miscellaneous:

4) i) 4:5 ii) 4:5 iii) 4:5 5) x = 62/3cm and y = 3.75 cm

6) 50m 30m 60m i) 12.5m2 ii) 720cm

3 9) i) 3:5 ii) 3:5 iii) 9:25

10) xy = 2.25cm i) 16x cm2 ii) 15x cm

2 11) i) 8 km ii) 30.72 km

2

14) scale factor = 1.5, DE = 4.5 cm and DF = 6 cm 15) 4:5

16) i) 4:25 ii) 4:9 17) 7.5 cm

cessf l u ut · 2019-02-24 · latest icse board question paper [2017–18] ... 144 ut sample paper...

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