circles and constructions - nims dubai · ms. seema lakra, s o ms. preeti hans ... circles and...

CBSE-i CBSE-i

Shiksha Kendra, 2, Community Centre, Preet Vihar, Delhi-110 092 India

(Core)

Circles and

Constructions

MATHEMATICS

CLASS - XUNIT-12

CBSE-i

UNIT-12

CLASS

X


Circles and

Constructions

(Core)

The CBSE-International is grateful for permission to reproduce

and/or translate copyright material used in this publication. The

acknowledgements have been included wherever appropriate and

sources from where the material may be taken are duly mentioned. In

case any thing has been missed out, the Board will be pleased to rectify

the error at the earliest possible opportunity.

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may be reproduced, printed or transmitted in any form without the

prior permission of the CBSE-i. This material is meant for the use of

schools who are a part of the CBSE-International only.

The Curriculum initiated by Central Board of Secondary Education -International (CBSE-i) is a progressive step in making the educational content and methodology more sensitive and responsive to the global needs. It signifies the emergence of a fresh thought process in imparting a curriculum which would restore the independence of the learner to pursue the learning process in harmony with the existing personal, social and cultural ethos.

The Central Board of Secondary Education has been providing support to the academic needs of the learners worldwide. It has about 11500 schools affiliated to it and over 158 schools situated in more than 23 countries. The Board has always been conscious of the varying needs of the learners in countries abroad and has been working towards contextualizing certain elements of the learning process to the physical, geographical, social and cultural environment in which they are engaged. The International Curriculum being designed by CBSE-i, has been visualized and developed with these requirements in view.

The nucleus of the entire process of constructing the curricular structure is the learner. The objective of the curriculum is to nurture the independence of the learner, given the fact that every learner is unique. The learner has to understand, appreciate, protect and build on values, beliefs and traditional wisdom, make the necessary modifications, improvisations and additions wherever and whenever necessary.

The recent scientific and technological advances have thrown open the gateways of knowledge at an astonishing pace. The speed and methods of assimilating knowledge have put forth many challenges to the educators, forcing them to rethink their approaches for knowledge processing by their learners. In this context, it has become imperative for them to incorporate those skills which will enable the young learners to become 'life long learners'. The ability to stay current, to upgrade skills with emerging technologies, to understand the nuances involved in change management and the relevant life skills have to be a part of the learning domains of the global learners. The CBSE-i curriculum has taken cognizance of these requirements.

The CBSE-i aims to carry forward the basic strength of the Indian system of education while promoting critical and creative thinking skills, effective communication skills, interpersonal and collaborative skills along with information and media skills. There is an inbuilt flexibility in the curriculum, as it provides a foundation and an extension curriculum, in all subject areas to cater to the different pace of learners.

The CBSE has introduced the CBSE-i curriculum in schools affiliated to CBSE at the international level in 2010 and is now introducing it to other affiliated schools who meet the requirements for introducing this curriculum. The focus of CBSE-i is to ensure that the learner is stress-free and committed to active learning. The learner would be evaluated on a continuous and comprehensive basis consequent to the mutual interactions between the teacher and the learner. There are some non-evaluative components in the curriculum which would be commented upon by the teachers and the school. The objective of this part or the core of the curriculum is to scaffold the learning experiences and to relate tacit knowledge with formal knowledge. This would involve trans-disciplinary linkages that would form the core of the learning process. Perspectives, SEWA (Social Empowerment through Work and Action), Life Skills and Research would be the constituents of this 'Core'. The Core skills are the most significant aspects of a learner's holistic growth and learning curve.

The International Curriculum has been designed keeping in view the foundations of the National Curricular Framework (NCF 2005) NCERT and the experience gathered by the Board over the last seven decades in imparting effective learning to millions of learners, many of whom are now global citizens.

The Board does not interpret this development as an alternative to other curricula existing at the international level, but as an exercise in providing the much needed Indian leadership for global education at the school level. The International Curriculum would evolve on its own, building on learning experiences inside the classroom over a period of time. The Board while addressing the issues of empowerment with the help of the schools' administering this system strongly recommends that practicing teachers become skillful learners on their own and also transfer their learning experiences to their peers through the interactive platforms provided by the Board.

I profusely thank Shri G. Balasubramanian, former Director (Academics), CBSE, Ms. Abha Adams and her team and Dr. Sadhana Parashar, Head (Innovations and Research) CBSE along with other Education Officers involved in the development and implementation of this material.

The CBSE-i website has already started enabling all stakeholders to participate in this initiative through the discussion forums provided on the portal. Any further suggestions are welcome.

Vineet Joshi

Chairman

PREFACEPREFACE

ACKNOWLEDGEMENTSACKNOWLEDGEMENTSAdvisory Conceptual Framework

Ideators

Shri Vineet Joshi, Chairman, CBSE Shri G. Balasubramanian, Former Director (Acad), CBSE

Sh. N. Nagaraju, Director(Academic), CBSE Ms. Abha Adams, Consultant, Step-by-Step School, Noida

Dr. Sadhana Parashar, Director (Training),CBSE

Ms. Aditi Misra Ms. Anuradha Sen Ms. Jaishree Srivastava Dr. Rajesh Hassija

Ms. Amita Mishra Ms. Archana Sagar Dr. Kamla Menon Ms. Rupa Chakravarty

Ms. Anita Sharma Ms. Geeta Varshney Dr. Meena Dhami Ms. Sarita Manuja

Ms. Anita Makkar Ms. Guneet Ohri Ms. Neelima Sharma Ms. Himani Asija

Dr. Anju Srivastava Dr. Indu Khetrapal Dr. N. K. Sehgal Dr. Uma Chaudhry

Coordinators:

Dr. Sadhana Parashar, Ms. Sugandh Sharma, Dr. Srijata Das, Dr. Rashmi Sethi, Head (I and R) E O (Com) E O (Maths) E O (Science)

Shri R. P. Sharma, Consultant Ms. Ritu Narang, RO (Innovation) Ms. Sindhu Saxena, R O (Tech) Shri Al Hilal Ahmed, AEO

Ms. Seema Lakra, S O Ms. Preeti Hans, Proof Reader

Material Production Group: Classes I-V

Dr. Indu Khetarpal Ms. Rupa Chakravarty Ms. Anita Makkar Ms. Nandita Mathur

Ms. Vandana Kumar Ms. Anuradha Mathur Ms. Kalpana Mattoo Ms. Seema Chowdhary

Ms. Anju Chauhan Ms. Savinder Kaur Rooprai Ms. Monika Thakur Ms. Ruba Chakarvarty

Ms. Deepti Verma Ms. Seema Choudhary Mr. Bijo Thomas Ms. Mahua Bhattacharya

Ms. Ritu Batra Ms. Kalyani Voleti

English :

Geography:

Ms. Sarita Manuja

Ms. Renu Anand

Ms. Gayatri Khanna

Ms. P. Rajeshwary

Ms. Neha Sharma

Ms. Sarabjit Kaur

Ms. Ruchika Sachdev

Ms. Deepa Kapoor

Ms. Bharti Dave Ms. Bhagirathi

Ms. Archana Sagar

Ms. Manjari Rattan

Mathematics :

Political Science:

Dr. K.P. Chinda

Mr. J.C. Nijhawan

Ms. Rashmi Kathuria

Ms. Reemu Verma

Dr. Ram Avtar

Mr. Mahendra Shankar

Ms. Sharmila Bakshi

Ms. Archana Soni

Ms. Srilekha

Science :

Economics:

Ms. Charu Maini

Ms. S. Anjum

Ms. Meenambika Menon

Ms. Novita Chopra

Ms. Neeta Rastogi

Ms. Pooja Sareen

Ms. Mridula Pant

Mr. Pankaj Bhanwani

Ms. Ambica Gulati

History :

Ms. Jayshree Srivastava

Ms. M. Bose

Ms. A. Venkatachalam

Ms. Smita Bhattacharya

Material Production Groups: Classes IX-X

English :

Ms. Rachna Pandit

Ms. Neha Sharma

Ms. Sonia Jain

Ms. Dipinder Kaur

Ms. Sarita Ahuja

Science :

Dr. Meena Dhami

Mr. Saroj Kumar

Ms. Rashmi Ramsinghaney

Ms. Seema kapoor

Ms. Priyanka Sen

Dr. Kavita Khanna

Ms. Keya Gupta

Mathematics :

Political Science:

Ms. Seema Rawat

Ms. N. Vidya

Ms. Mamta Goyal

Ms. Chhavi Raheja

Ms. Kanu Chopra

Ms. Shilpi Anand

Geography:

History :

Ms. Suparna Sharma

Ms. Leela Grewal

Ms. Leeza Dutta

Ms. Kalpana Pant

Material Production Groups: Classes VI-VIII

1. Syllabus 1

2. Scope document 2

3. Teacher's Support Material 4

Teacher Note 5

Activity Skill Matrix 10

Warm Up W1 11

Recalling Circle

Warm Up 2 11

Vocabulary of circles

Pre -Content P1 13

Circle Passing through three points

Content Worksheet CW1 14

Radius and Tangent


Length of Tangents


Constructing Tangents

Post Content Worksheet PCW1 19




Assessment Plan 19

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Content

4. Study Material 23

5. Student's Support Material 47

SW1: Warm Up (W1) 48

Recalling Circle

SW2: Warm Up (W2) 50


SW3: Pre Content (P1) 56


SW4: Content (CW1) 58

Radius and Tangent


Length of Tangents



SW7: Post Content (PCW1) 82

SW8: Post Content (PCW2) 84

SW 9: Post Content (PCW3) 86

SW 10: Post Content (PCW4) 87

6. Suggested Videos & Extra Readings. 89

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1

SYLLABUS

Circles and Constructions (Core)

Introduction

Circle and line

Recall basic terms and results

Tangent and secant

Theorems through self exploration and proof

Theorem:

1) Tangent at any point of a circle is perpendicular to the radius through the point of contact.

2) The lengths of tangents drawn from an

external point to a circle are equal.

Constructions Construction of tangents to a circle from an external point

2

SCOPE DOCUMENT

Key concepts

1. Point of contact

2. Tangent

3. Secant

4. Chord

5. Perpendicular bisector

6. Length of tangent

Learning objectives:

Understand all terms related to circle : chord, tangent, secant etc.

Verify that tangent at any point of a circle is perpendicular to the radius through

the point of contact.

Understand the proof of theorem that tangent at any point of a circle is

perpendicular to the radius through the point of contact.

Solve the problems based on theorem that tangent at any point of a circle is


Verify that the lengths of tangent drawn from an external point to a circle are

equal.

Understand the proof of the theorem that the lengths of tangent drawn from an

external point to a circle are equal.

Solve the problems based on the theorem that the lengths of tangent drawn from

an external point to a circle are equal.

3

Construct the tangents to a circle drawn from an external point.

Extension Activities:

1. Given a quadrilateral inscribed in a circle, how will you find the center and radius of

the circle?

2. Given a circle inscribed in a quadrilateral, how will you find the center and radius of

the circle?

Research:

Explore the relation between a triangle and a circle and try to find the answer to

following questions:

A) How many cases are possible when there is 0 points of intersection?

B) How many cases are possible when there is 1 points of intersection?

C) How many cases are possible when there are 2 points of intersection?

D) How many cases are possible when there are 3 points of intersection?

E) How many cases are possible when there are 4 points of intersection?

F) How many cases are possible when there are 5 points of intersection?

G) How many cases are possible when there are 6 points of intersection?

H) CROSS-CURRICULAR LINK:

A potter found a piece of beautiful circular plate. He wants to produce replica of

original size. How can he determine the original size of the plate?

4

Teacher’s

Support

Material

5

TEACHER’S NOTE The teaching of Mathematics should enhance the child’s resources to think and reason,

to visualize and handle abstractions, to formulate and solve problems. As per NCF

2005, the vision for school Mathematics includes:

1. Children learn to enjoy mathematics rather than fear it.

2. Children see mathematics as something to talk about, to communicate through, to

discuss among them, to work together on.

3. Children pose and solve meaningful problems.

4. Children use abstractions to perceive relationships, to see structures, to reason out

things, to argue the truth or falsity of statements.

5. Children understand the basic structure of Mathematics: Arithmetic, algebra,

geometry and trigonometry, the basic content areas of school Mathematics, all

offer a methodology for abstraction, structuration and generalisation.

6. Teachers engage every child in class with the conviction that everyone can learn

mathematics.

Students should be encouraged to solve problems through different methods like

abstraction, quantification, analogy, case analysis, reduction to simpler situations, even

guess-and-verify exercises during different stages of school. This will enrich the

students and help them to understand that a problem can be approached by a variety of

methods for solving it. School mathematics should also play an important role in

developing the useful skill of estimation of quantities and approximating solutions.

Development of visualisation and representations skills should be integral to

Mathematics teaching. There is also a need to make connections between Mathematics

and other subjects of study. When children learn to draw a graph, they should be

encouraged to perceive the importance of graph in the teaching of Science, Social

6

Science and other areas of study. Mathematics should help in developing the reasoning

skills of students. Proof is a process which encourages systematic way of

argumentation. The aim should be to develop arguments, to evaluate arguments, to

make conjunctures and understand that there are various methods of reasoning.

Students should be made to understand that mathematical communication is precise,

employs unambiguous use of language and rigour in formulation. Children should be

encouraged to appreciate its significance.

At the secondary stage students begin to perceive the structure of Mathematics as a

discipline. By this stage they should become familiar with the characteristics of

Mathematical communications, various terms and concepts, the use of symbols,

precision of language and systematic arguments in proving the proposition. At this

stage a student should be able to integrate the many concepts and skills that he/she has

learnt in solving problems.

The unit on circle and construction is built upon the knowledge gained in previous

classes and introduce the students to new concepts of secants and tangents. In this unit

the focus is on hands on activities to give concrete experience to learners so that they

can frame the generalized statement of theorem on their own.

The learning objectives in this unit are:

Understand all terms related to circle- chord, tangent, secant etc.

Verify that tangent at any point of a circle is perpendicular to the radius

through the point of contact.

Understand the proof of theorem that tangent at any point of a circle is


Solve the problems based on theorem that tangent at any point of a circle is


7

Verify that the lengths of tangent drawn from an external point to a circle are

equal.

Understand the proof of the theorem that the lengths of tangent drawn from

an external point to a circle are equal.

Solve the problems based on the theorem that the lengths of tangent drawn

from an external point to a circle are equal.

Construct the tangents to a circle drawn from an external point.

Warm up activity W1 will help the students to recall or to explore all possible relations

between a line and circle and an arc and a circle. This activity can also be used to

introduce the term like chord, secant, target, sector, arc, segment etc. The activity also

help the students to coin the definition of all these terms on their own. Activity W2 will

further Strengthen their understanding of definitions of all circle related terms.

Pre-content activity P1 will allow the students to visualize that through three non-

collinear points only one circle can pass through. They shall be guided by teacher to

make this observation using their knowledge of perpendicular bisector as well the

properties of Chords of circle. During this activity teacher can be a facilitator instead of

being instructor. Learners can be given freedom of exploration here.

Warm up activity W1 and W2 are to be followed by content worksheet CW1 to reinforce

the circle related concepts and all terms. To clarify the concepts activities are suggested

of both Warm up level as well as content transaction level. This repetition help all type

of learners to grasp the basic ideas. Exploration of Theorems as suggested in task 2 and

Task 3 will help the students to observe and to make conjectures and task 4 will provide

opportunity to make use of the theorems in problems. Same procedure is adopted for

understanding of all theorems in further content worksheets. Teachers should take care

of exploring each theorem logically as the exploration of theorem is done through 5-7

examples only. Learner should not take this impression that any observation common

8

with 5-7 cases is true for all cases. Before generalizing any observation logical

verification is must. Although the proof of theorems may not be asked in examination

from students of ‘Love Mathematics’.

Theorem on length of tangents from external point can be demonstrated in a very

interesting manner. When two lines are drawn at certain angle, several discs of different

radii / coins can be placed between two lines to demonstrate that length of tangent

drawn from an external point are always equal.

To develop the students’ understanding following questions can be posed to them:

a) Is it possible to construct a single circle which has both rays as tangents?

b) (Once they are able to place more than one circular disc as shown above)

Find the diameter of each disc and construct the centre of each circle.

c) What is the relation between the centre of the circle and an angle whose rays are

both tangent to the circle?

d) What is the name given to the distance between the vertex and the point of contact?

e) What is the angle formed between radius and the point of contact?

f) How can you determine the centre of a circle inscribed in a triangle?

g) How can you determine the centre of a circle inscribed in a square?

9

This activity will help the learners to internalize all concepts and theorems clearly.

Each theorem is followed by the problem sheet containing lots of riders based on it.

Construction of length of tangents is first explained with paper folding and then with

the help of geometrical instruments.

10

ACTIVITY SKILL MATRIX

Type of Activity Name of Activity Skill to be developed

Warm UP(W1) Recalling Circles Thinking ,relating to learnt concepts

Warm Up (W2) Vocabulary of circles

Expression,

Pre-Content (P1) Circle Passing through three points

Observation, inferential, critical thinking, problem solving

Content (CW 1) Radius and tangent

Observation, inferential, critical thinking, problem solving

Content (CW 2) Length of tangent Observation, inferential, critical thinking, problem solving

Content (CW 3) Constructing tangents

Construction, precision

Post - Content (PCW 1)

Problem solving skills.


Problem solving skills.


Conceptual knowledge.


Knowledge and application.

11

ACTIVITY1- WARM UP W1

Recalling Circle

Specific Objective:

To recall the terms related to circles.

Description: Students will recall as many terms as they can after looking at the pictures

given in the task.

Execution: Teacher may show the pictures on projector and ask the students to recall

all the terms related to circles.

Parameters for Assessment:

Able to recall basic terms related to circle.

ACTIVITY2- WARM UP W2


Specific Objective:

To revisit basic terms related to circle.

Description: This is a recall activity. Students will cut the given 8 pieces of the puzzle

and will arrange them to form a circle such that the term and its definition run parallel

with the top and bottom edges of the segment after joining.

Execution: Teacher may distribute the puzzle sheet to each student. They will cut the

pieces, arrange them on a chart paper and paste them to form a circle.

12

Solution Sheet:


Able to recall definitions of basic terms related to circle.

13

ACTIVITY3- PRE CONTENT (P1)


Specific Objective:

To recall construction of a circle passing through 3 points.

Description: Students have already learned in their earlier classes how to construct a

circle through 3 given points. This task is a recall exercise for them.

Execution: Teacher may describe the situation to the students and may show the

pictures with the help of projector. Let children recall the steps of construction of circle

through 3 given points. A class discussion should be organized thereafter to discuss the

steps which may have been done by Romil.

Students may be encouraged to give the justification of the steps of construction.


Able to recall perpendicular bisector of a line segment

Able to recall properties of chords of a circle.

Able to recall the construction of circle through 3 points.

Extra Reading:

http://www.mathopenref.com/const3pointcircle.html


14

ACTIVITY 4- CONTENT (CW1)

Radius and Tangent

Specific Objective:

To understand tangents and related terms.

To verify that tangent is perpendicular to the radius.

Description: Task 1 is a word search game. Students will search the terms learnt in this

unit and will encircle them. Further, they will explain each term with the help of

diagrams.

In task 2, students will measure the angle between the radius and the tangent with the

help of protractor to verify that radius is perpendicular to the tangent at the point of

contact.

Task 3 is an exploratory task to find the relationship between parallel tangents and the

diameter.

Task 4 contains questions based on the concepts learnt.

Execution: Printed worksheets may be distributed for task 1. Each student will solve the

worksheet and may paste it in their notebooks after completing.

15

Solution grid:

For task 2, teacher may distribute printed sheets. Each student will measure the angle

with the help of protractor. Students will write their reflections in the sheets thereafter.

Teacher may explain task 3 and students may do hands on activity to find the

relationship.

Teacher may distribute printed worksheets for task 4. Alternatively, teacher may write

the questions on board and students may solve these in their notebooks.

A class discussion may follow to clear the doubts of students.


Able to recall basic terms such as tangent, secant and point of contact

16

Able to verify that radius is perpendicular to the tangent.

Able to apply Pythagoras theorem to solve questions

Able to solve problems based on the relation between tangent and radius.

Extra Reading:

http://www.youtube.com/watch?v=Ut_rKPch-JE&feature=player_embedded

http://bcove.me/axis2i2q

http://www.youtube.com/watch?v=FmXxPMFifSs&feature=player_embedded

http://www.youtube.com/watch?v=E2uoEMwuyak&feature=player_embedded





17


Length of Tangents

Specific Objective:

To verify that length of tangents from an external point are equal.

Description: In task 1, students will measure the length of tangents with the help of

ruler to verify that the lengths of tangents from an external point to a circle are equal.

Task 2 is an exploratory task which gives students a chance to verify the results with

hands on activity.

Task 3 also contains problems based on the length of tangents.

Execution: For task 1, teacher may distribute printed sheets. Each student will measure

the length of tangents with the help of ruler and will write it in the space provided.

Students will write their reflections in the sheets thereafter.

Teacher may explain task 3 and students may do hands on activity to find the

relationship.

Teacher may take task 3 as a black board task. Students can solve the problems on

board one by one. They may do the problems individually thereafter in their notebooks.


Able to understand the concept of length of tangents.

Able to verify that length of tangents from an external point to a circle are equal

Able to solve problems based on length of tangents.

Extra Reading:

http://bcove.me/t8tsw680


18



Specific Objective:

To construct tangents to a circle using ruler and compass.

Description: Task 1 is a hands on activity. Students will use paper folding to make

tangents at given points. For this they need to recall that tangent is perpendicular to the

radius at the point of contact. They may draw lines passing through the given points

using paper folding and then they may make perpendiculars to these lines at given

points by paper folding.

Task 2 contains questions involving construction of tangents.

Execution: Teacher may distribute printed worksheet for task 1. Students will make

tangents on given points with paper folding. They may mark the folds or the crease by

dotted lines. Students may paste this sheet in their notebooks and should write the steps

used and justify these steps in their notebooks.

Teacher may write the questions in task 2 on black board. Students may do these

questions in their notebooks.


Able to construct perpendicular bisector of a given line segment

Able to construct tangents to a given circle from an external point.

Able to find the centre of a given circle

Extra Reading:

http://www.mathopenref.com/consttangent.html

http://www.mathopenref.com/constcirclecenter.html


http://www.mathopenref.com/consttangents.html





19

ACTIVITY 7- POST CONTENT (PCW1)

Students will be assessed on the worksheet containing questions based on tangents to a

circle.


Students will be assessed on the worksheet containing questions based on length of

tangents.

ACTIVITY 9-POST CONTENT (PCW3)

Students will be assessed on the worksheet containing questions based on construction

of tangents to a circle.


Assessment of the students will be done by using word puzzle.

ASSESSMENT PLAN

Assessment guidance plan for teachers

With each task in student support material a self –assessment rubric is attached for

students. Discuss with the students how each rubric can help them to keep in tune their

own progress. These rubrics are meant to develop the learner as the self motivated

learner.

To assess the students’ progress by teacher two types of rubrics are suggested below,

one is for formative assessment and the other is for summative assessment.

20

SUGGESTIVE RUBRIC FOR FORMATIVE ASSESSMENT

(EXEMPLARY)

Parameter Mastered Developing Needs motivation

Needs personal attention

Theorems related to circle

Can state the

theorems

correctly

Can identify

the theorem

applicable in

given figure

Can prove all

theorems

Can solve the

problems based

on theorems

Can state the

theorems

correctly

Can identify

the theorem

applicable in

given figure

Can prove all

theorems

Can not solve the problems based on theorems

Can state the

theorems

correctly

Can identify

the theorem

applicable in

given figure

Can not

prove all

theorems


Can not state

the theorems

correctly

Can not

identify the

theorem

applicable in

given figure

Can not

prove all

theorems


From above rubric it is very clear that

Learner requiring personal attention is poor in concepts and requires the training

of basic concepts before moving further.

Learner requiring motivation has basic concepts but may have problem in

drawing the figure or in writing the steps logically.

Learner who is developing may be able to write the steps of proof intuitively but

may not know the logic or vice-versa.

Learner who has mastered has acquired all types of skills that are required to

solve the problems.

21

TEACHERS’ RUBRIC FOR

SUMMATIVE ASSESSMENT OF THE UNIT

Parameter 5 4 3 2 1

Circle and line

Able to define and draw figure of tangent and secant of a circle correctly.

Able to define all

terms related to circle

Not able to define and draw figure of tangent and secant of a circle correctly

Not able to define the terms related to circle.

Theorems Able to explain the statement of the theorem-

1) Tangent at any point

of a circle is perpendicular to the radius through the point of contact.

2) The length of tangents drawn from an external point to a circle are equal.

Able to verify the above stated theorems

Able to solve the problems based on the theorems stated above

Not able to explain the statement of the theorem-

1) Tangent at any

point of a circle is perpendicular to the radius through the point of contact.

2) The length of tangents drawn from an external point to a circle are equal.

Not able to verify

the above stated theorems

Not able to solve the problems based on the theorems stated above

22

Construction

Able to construct tangents to a circle from an external point accurately and neatly

Not able to construct tangents to a circle from an external point accurately and neatly

23

Study

Material

24

CIRCLES AND CONSTRUCTIONS (Core)

INTRODUCTION

In Class IX, we have discussed a circle and basic terms such as centre, radius, arc, chord

etc, related to a circle and some important properties. In this unit, we shall first briefly

recall these basic concepts and properties and then extend this knowledge to learn the

concept of a tangent to a circle and some properties of the tangent. Finally, we shall also

discuss the construction of a tangent to a circle from an external point.

1. Basic Terms and Results related to Circle: A Recall

Circle: A circle is a collection of all points in a plane, which are at a fixed distance

from a fixed point in the plane.

Fixed point is called the centre and the fixed distance is called the radius.

In Fig.1, O is the centre and OP is the radius of the circle.

Fig. 1

25

Chord of a circle

If P and Q are any two points on a circle, then the line segment PQ is called a chord of the circle.

Fig.2

In Fig. 2, PQ is a chord of the circle.

Diameter of a circle

The chord passing through the centre of the circle is called a diameter of the circle. In Fig. 2, AB is a diameter. Clearly a diameter is the longest chord.

Arc of a circle

A part of circle between two points on the circle is called an arc of the circle.

In general, there are two parts one longer and the other smaller. Longer part is called the major arc and smaller part is called the minor arc.

In Fig. 3, APB is a major arc and AQB is a minor arc

Fig. 3

26

When the two parts (arcs) are equal, each is called a semicircle.

In Fig. 4, APB and AQB are two semicircles.

Fig. 4

Segment of a circle

Recall that a circle divides a plane into three parts:

(i) Interior

(ii) Exterior

(iii) Circle itself

(See Fig.5)

Fig. 5

Circle along with its interior is called the circular region

A chord of a circle divides the circular region into two parts. Each part is called a segment of the circle.

In general, one segment is bigger.

27

The bigger one is called the major segment and the smaller one is called the minor

segment of the circle.

In Fig. 6, APB is the major segment and AQB is the minor segment.

Fig. 6

Cyclic Quadrilateral

A quadrilateral whose all the four vertices lie on a circle, is called a cyclic

quadrilateral.

In Fig. 7, ABCD is a cyclic quadrilateral.

Fig. 7

Some Basic Results

(i) Two circles are congruent if they have equal radii.

(ii) Equal chords of a circle (or of congruent circles) subtend equal angles at the

centre(s) and its converse.

(iii) If two arcs of a circle (or of congruent circles) are equal then their

corresponding chords are equal and its converse.

28

(iv) If two arcs of a circle (or of congruent circles) arc congruent, then the angles

subtended by them at the centre (s) are equal.

(v) The perpendicular from the centre of a circle to a chord bisects the chord.

Conversely, a line drawn through the centre of a circle to bisect the chord is

perpendicular to the chord.

(vi) There is one and only one circle passing through three non collinear points.

(vii) Equal chords of a circle (or of congruent circles) are equidistant from the

centre(s) and its converse.

(vii) The angle subtended by an arc of a circle at the centre is double the angle at

any point on the remaining part of the circle.

(viii) Angle in a semicircle is a right angle and conversely a circle drawn on the

hypotenuse of a right triangle as a diameter passes through the opposite

vertex.

(ix) Angles in the same segment of a circle are equal and its converse.

(x) Opposite angles of a cyclic quadrilateral are supplementary and its converse.

2. Secant and Tangent

Consider a circle with centre O and a line AB in the plane of the circle. This line can have three positions relative to the circle as shown below:

Fig. 8

29

In Fig. 8 (i), line AB does not intersect the circle.

In Fig. 8 (ii) line AB intersects the circle at two district points P and Q. We say that

AB is a secant of the circle. Recall that chord PQ is a part of the secant AB.

In Fig. 8(iii), line AB intersects the circle at two coincident points. We also say that

line AB intersects the circle at only one point or touches the circle at a point P, or

AB is a tangent to a circle at the point P.

Point P is called the point of contact.

Thus,

A tangent to a circle is a line which intersects the circle in exactly one point. Clearly,

at a point of a circle, there is one and only one tangent.

A tangent can be considered as a limiting case of a secant when two points of

intersection of a secant with the circle coincide each other.

(See Fig. 9)

Fig. 9

3. Number of Tangents to a Circle from a Point

(i) Let P be a point inside the circle. The circle.

Try to draw a tangent from P to the circle.!!

30

You will see that no tangent to the circle can be drawn through a point lying inside the circle

Fig. 10

(ii) Let P be a point on the circle.

Try to draw a tangent to the circle at the point P

You will see that one and only one tangent to a circle from a point lying on the circle. (Fig.11)

Fig. 11

(iii) Let P be a point outside the circle. Try to draw a tangent from P to the circle!!

31

You will see that two tangents can be drawn from a point lying outside the circle. (Fig. 12)

Fig. 12

Example 1: Fill in the blanks:

(i) A secant to a circle intersects it in ________ points

(ii) Minimum number of points of intersection of a line and a circle is ________ and maximum number of points of intersection of a line and a circle is ______

(iii) A tangent to a circle intersects it in ______ point(s)

(iv) A line intersecting a circle in two district points is called a_________

(v) The common point of a tangent to a circle and the circle is called ______

(vi) From an external point of a circle, ________ tangents can be drawn. From a point inside the circle, ______ tangent (s) can be drawn.

Solution:

(i) Two

(ii) Zero; two

(iii) One

(iv) Secant

(v) Point of contact

(vi) Two; No

32

4. Tangent and Radius

Let OP be a radius of a circle with centre O.

Through P, draw a line AB intersecting the circle at only one point P.

Fig. 13

Measure OPA and OPB.

You will find that OPA = OPB = 90o.

Thus, OP AB at P.

Or, radius of a circle is perpendicular to the tangent at the point of contact P.

In other words, the tangent at any point of a circle is perpendicular to the radius

through the point of contact

We can also prove it in the form of a theorem as follows:

Theorem 1: The tangent at any point of a circle is perpendicular to the radius

through the point of contact.

Given: A circle with centre O and a tangent AB to the circle at the point P

(See Fig. 14)

To Prove: OP AB

Construction: Take any point Q other than P on AB and join OQ.

33

Proof: The point Q cannot lie inside the circle as no tangent can be drawn through a

point inside the circle.

Fig. 14

So, Q must lie outside the circle.

Thus, OQ > OP (radius)

Since Q can be any point on the tangent AB, so, for every position of point Q, OQ >

OP

i.e. of all the line segment drawn from O to tangent AB, line segment OP is the

shortest.

Thus, OP AB.

Hence, proved.

5. Tangents from an External Point

Take a circle with centre O and also take a point P outside the circle. Through P,

draw lines touching the circle at only one point.

34

PA and PB are such two lines. Measure the line segment PA and PB.

Fig. 15

You will find that PA = PB.

i.e., the lengths of the line segments corresponding to two tangents drawn from P

are equal.

We also say that length of two tangents from an external point P are equal.

We can also prove this result as follows:

Theorem 2: Lengths of two tangents drawn from an external point to a circle are equal.

Given: A circle with centre O and a point P lying outside the circle. PA and PB are two

tangents. (See Fig. 16).

Fig. 16

Construction: Join OP, OA and OB.

Proof: In OPA nad OPB,

35

OP = OP (Common)

OA = OB (Radii of a circle)

PAO = PBO (Each equal to 90o as radii and tangents are perpendicular)

So, OPA OPB (RHS)

Therefore, PA = PB (CPCT)

Hence proved.

Note: Since OPA OPB, therefore, APO = BPO

i.e., APO = APB and BPO = APB, i.e., the line OP joining the centre to

the external point bisects the angle between the two tangents.

Let us take some examples to illustrate applications of these theorems.

Example 2: Prove that the tangents drawn at the ends of a diameter of a circle are

parallel.

Solution: PQ and RS are two tangents drawn at A and B respectively at the ends of the

diameter of a circle with centre O.

Fig. 17

36

OA PQ

So, OAQ =

Similarly, OBS =

Now OAQ + OBS = + = 1

i.e., the sum of interior angles on the same side of the transversal AB is 180o.

So, PQ RS.

Example 3: From a point T, the length of the tangent to a circle, with centre O, is 12cm

and the distance of the point T from the centre is 12.5 cm. Find the radius of the circle.

Solution: Let OP = x cm (Fig. 18)

Fig. 18

OPT = [OP PT]

OT2 = OP2 + PT2 [Using Pythagoras Theorem in right triangle OPT]

(12.5)2 = x2 + 122

x2 = (12.5)2 – (12)2

= (12.5 + 12) (12.5 – 12) = (24.5) (.5)

= 12. 25

x = 3.5

37

So, radius of circle = 3.5 cm

Example 4: In the figure, TP and TQ are tangents from a point T to the circle with centre

O. If POQ = 1 , find PTQ.

Solution: OPT = OQT = [As TP and TQ are tangents]

In quadrilateral POQT,

Fig. 19

OPT + PTQ + OQT + POQ = 3 [Angle sum property of a quadrilateral]

+ PTQ + + 11 = 3

PTQ = 36 – 11 – 18

= 7

Note: In general, if TP and TQ are tangents from an external point T to the circle with

centre O, then POQ and PTQ are supplementary

38

Example 5: A quadrilateral ABCD circumscribes a circle as shown in Fig. 20.

Prove that AB + CD = AD + BC.

Fig. 20

Solution:

nal points)

DR = DS

AP = AS(Tangents to a circle from exter

BP = BQ

and CR = CQ

Adding, we get

DR + AP + BP + CR = DS + AS + BQ + CQ

(DR + CR) + (AP + PB) = (DS + SA) + (BQ + QC)

CD + AB = DA + BC

Example 6: Prove that the opposite sides of a quadrilateral circumscribing a circle

subtend supplementary angles at the centre of the circle.

39

Solution: Let ABCD be the quadrilateral which circumscribes a circle with centre O (See

Fig. 21).

Fig. 21

We are to prove that

AOB + COD = 18

and AOD + BOC = 18

APO = 9 (Angle between radius and tangent)

Also PAO = A

So, POA = 18 – 9 – A

= 9 – A (1)

Similarly POB = 9 – B, (2)

ROC = 9 – C, (3)

and ROD = 9 – D, (4)

40

Adding (1), (2), (3) and (4), we get

POA + POB + POC + POD = 9 – A + 9 – + 9 – C + 9 – D

AOB + COD = 36 – A + B + C + D)

= 36 – x 360o

= 36 – 18 = 18

Similarly, it can be proved that

AOD + BOC = 180o

6. Construction: To construct the tangents to a circle from a point outside it.

We explain the construction through an example.

Example 7: Draw a circle of radius 3cm and from a point 5cm away from its centre,

construct the tangents to the circle.

Solution: We go through the following steps:

1. With O as centre, draw a circle of radius 3cm.

2. Mark a point P such that OP = 5cm.

3. With OP as diameter, draw a circle intersecting the first circle at points Q and R.

4. Join PQ and PR.

5. Then, PQ and PR are the required tangents to the circle from the external point P.

(Fig. 22)

41

Fig. 22

Example 8: Construct a pair of tangents to a circle of radius 6cm which arc inclined to

each other at an angel 60o.

Solution: We go through the following steps:

1. Draw a circle of radius 6 cm with centre O.

2. Draw two radii OA and OB of the circle such that AOB = 18 – 6 = 12

3. Draw a line OA and a line m OB to intersect each other at the point P.

Then, PA and PB are the required two tangents inclined to each other at an

angle of 6 .

(See Fig. 23)

Fig. 23

42

6. Angles in the Alternate Segment

Consider a circle with centre O.

Let XAY be a tangent to the circle at the point A.

Draw a chord AB making two angles with the tangent, namely BAY and BAX.

Chord AB divides the circle into two segments ADB and ACB (See Fig. 24).

The segments ADB and ACB are called alternate segments of angles BAY and

BAX respectively.

There is a result known as Alternate Segment Theorem which relates angles between a

tangent and a chord (with its one end point as the point of contact) and the angles

formed in their respective alternate segments.

Fig. 24

Theorem 3: If a line touches a circle and from the point of contract a chord is drawn,

the angles which this chord makes with given line (tangent) are equal respectively to

line the angles formed in the respective alternate segments.

Given: PQ is a tangent to a circle with centre O at a point A.

AB is a chord and C and D are points in the two segments of the circle formed by AB

(Fig. 25).

43

Fig. 25

To Prove: BAQ = ACB

and BAP = ADB

Construction: Draw the diameter AOE, and join EB.

Proof: In AEB, ABE = 9 (Angle in a semi circle)

So, AEB + EAB = 9 (1)

Since EA PQ, therefore

EAB + BAQ = EAQ = 9 (2)

From (1) and (2),

AEB = BAQ

But ACB = AEB (Angles in the same segment)

So, BAQ = ACB

Similarly, it can be proved that

44

BAP = ADB.

Hence, proved.

The converse of the theorem is also true. It is stated as follows:

Theorem 4: If a line is drawn through an end point of a chord of a circle so that the

angle formed by it with the chord is equal to the angle subtended by the chord in the

alternate segment, then the line is a tangent to the circle.

This theorem can be proved by the method of contradiction as follows:

If PAQ is not a tangent, draw the tangent P’AQ’ to the circle at A.

(See Fig. 26). Now since P’AQ’ is a tangent at A,

BAQ’ = ACB (By Theorem 3) (1)

Also BAQ = ACB (Given) (2)

So, BAQ = BAQ’ [From (1) and (2)]

Fig. 26

Unless ray AQ’ coincide with AQ, this is impossible.

45

Thus, P’AQ’ coincides with PAQ or PAQ is the tangent to the circle at A.

Now, Let us consider some examples.

Example 9: In the figure, DAB is a tangent to the circle at A, RAB = 4 . Find RQA.

Fig. 27

Solution: RPA = RAB (Angles in alternate segments)

= 4

Also, RPA + RQA = 18 (opposite angles of a cyclic quadrilateral).

So, 4 + RQA = 18

Or RQA = 14

Example 10: ABC is a triangle right angled at B. A circle with the side AB as a diameter is drawn to intersect the hypotenuse AC in P. Prove that the tangent to the circle at P bisects the side BC.

Fig. 28

46

Solution: Let the tangent at P to the circle with AB as diameter intersect BC in Q.

We have to prove BQ = CQ. Join BP.

APB = 90o (Angle in a semicircle)

BPC = 9 ( APB and BPC for an linear pair)

Now BPC = BAC + BCA = 900 (1)

also BPQ + QPC = BPC = 900 (2)

from (1) and (2) we get

or, BPQ + QPC = BAC + BCA

Also, BPQ = BAC (angles in alternate segments)

So, QPC = BCA

Thus, PQ = QC

Also, PQ = QB (Tangents from an external point)

Hence, QC = QB

Hence proved.

Example 11: Two circles intersect at A and B. From a point P on one of these circles, two line segments PAC and PBD are drawn intersecting the other circle at C and D respectively. Prove that CD is parallel to tangent at P. (Fig. 29)

Fig. 29

Solution: Join AB.

APE = ABP (Angles in alternate segments) (i)

Also, ABD + ACD = 18 ……(ii) (angles of a cyclic quadrilateral)

47

and ABD + ABP = 18 ……(iii) (Linear pair)

ACD = ABP (from (ii) and (iii) (iv)

Therefore, ACD = APE (from (i) and (iv)

So, EPF CD (Alternate angles)

Hence, the result.

48

Student’s

Support

Material

49

STUDENT’S WORKSHEET 1

WARM UP (W1)

Recalling Circle

Name of Student___________ Date________

Given below is a line, an arc and a circle.

Write as many terms as you can use for the above three pictures.

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SELF ASSESSMENT RUBRIC 1 – WARM UP 1 (W1)

Parameter

Able to recall basic terms

related to circle.

51


WARM UP (W2)



Cut the following pieces. Arrange them in a form of circle such that the term and its

definition run parallel with the top and bottom edges of the segment after joining.

Solution format:

52

Puzzle Pieces:

56

SELF ASSESSMENT RUBRIC 2 – WARM UP (W2)

Parameter

Able to recall definitions

of basic terms related to

circle.

57


Pre Content (P1)



Ramesh wants to draw a circle that passes through 3 points A, B and C below.

Using your knowledge of circle, suggest some ways to draw a circle passing through all

the three points A, B, C.

How many such circles can you draw?

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SELF ASSESSMENT RUBRIC 3 – PRE CONTENT (PC1)

Parameter

Able to recall

perpendicular bisector of

a line segment

Able to recall properties

of chords of circle.

59


Content Worksheet (CW1)

Radius and Tangent


Task 1:

Given below is a word search game. Search the terms given on the right in the grid.

Use a stop watch to see how quick you are.

60

Now explain these terms with the help of diagrams.

Explanation of Terms Diagram

Tangent _____________________________ _____________________________ _____________________________ _____________________________ _____________________________ _____________________________

Secant _____________________________ _____________________________ _____________________________ _____________________________ _____________________________ _____________________________

Point of Contact _____________________________ _____________________________ _____________________________ _____________________________ _____________________________ _____________________________

Perpendicular Bisector _____________________________ _____________________________ _____________________________ _____________________________ _____________________________ _____________________________

Chord _____________________________ _____________________________ _____________________________ _____________________________ _____________________________ _____________________________

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Length of tangent _____________________________ _____________________________ _____________________________ _____________________________ _____________________________ _____________________________

Task 2:

Measure the angle between the radius and the tangent at the point of contact using

protractor, for each of the given figure.

Figure Measure of angle between radius and tangent

63

Based on your observation, comment on the following statement,

“The radius is always perpendicular to the tangent at the point of contact.”

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Task 3: Explore the relationship between the tangents drawn at the end points of a

diameter of a circle.

64

Task 4: Do the following

1. Determine if line AB is tangent to the circle.

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65

2. Find the segment length indicated. Assume that lines which appear to be

tangent are tangent.

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3. Find the angle measure indicated. Assume that lines which appear to be tangent

are tangent.

66

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4. From a point Q, the length of the tangent to a circle is 24 cm and the distance of

Q from the centre is 25 cm. Find the radius of the circle.

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5. Given circle O with segment AB tangent to the circle at A. If OA = AB, what kind of triangle is OAB?

a scalene triangle an isosceles right

triangle an equilateral

triangle an equilateral

acute triangle an isosceles

obtuse triangle

68

SELF ASSESSMENT RUBRIC 4

Content Worksheet (Cw1)

Parameter

Able to recall basic terms

such as tangent, secant

and point of contact

Able to verify that radius

is perpendicular to the

tangent.

Able to apply Pythagoras

theorem to solve

questions

Able to solve problems

based on the relation

between tangent and

radius.

69



Length of Tangents

Activity Skill Matrix- Circle and construction Task 1:

Measure the length of tangents from an external point with the help of ruler for each of

the given figure.

Figure Measure of tangent lengths.

Length of first tangent = …….. Length of second tangent = ……..


70




71

Generalize your observations.

Task 2: Given an angle ABC with rays AB and BC, look for circular objects( coins,

bangles, cut outs of circles) such that AB and BC becomes tangent to these circles.

Based on your observation, comment on the following statement,

“Length of tangents to a circle from an external point are always equal.”

Support your comments with proper justification.

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1. Find the value of x.

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2. Find the perimeter for each polygon.

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3. In the accompanying diagram, circle O is inscribed in triangle ABC so that AB, BC and AC are tangents to the circle at F, E and D respectively. If AF = FB = 5 and DC = 7, find the perimeter of triangle ABC.

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75

4. In the diagram below, PA and PB are tangents to circle with centre O. If PA = 10 and

OA = 5, find the perimeter of quadrilateral PAOB.

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76

5. In the diagram below, ST, QP, and SQ are tangents to circle O. If ST = 5 and QP = 2, find SQ.

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77



Parameter

Able to understand the

term length of tangents.

Able to verify that length

of tangents from an

external point to a circle

are equal

Able to solve problems

based on length of

tangents.

78


CONTENT WORKSHEET (CW3)



Task 1:

Draw a circle with centre O. Take points A, B, C and D on it.

Make tangents at A, B, C and D using paper folding. Use the fact that radius is

perpendicular to the tangent at the point of contact.

79


1. Construct tangent through an external point by following the steps of construction

given below.

Steps Of Construction Actual Construction

Given:

A circle with centre O and a point P in the exterior.

1. Draw a straight line between the center O of the given circle and the given point P.

2. Find the midpoint of this line by constructing the line's perpendicular bisector.

3. Place the compass on the midpoint just constructed, and set it's width to the center O of the circle.

4. Without changing the width, draw arcs across the circle in the two possible places. These are the contact points J, K

80

Steps Of Construction Actual Construction

for the tangents.

5. Join PJ and PK

6. PJ and PK are the required tangents from the given external point to the circle.

81

2. Find the center of the circle given below. Also draw a tangent to the circle. Write the steps of construction.

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82

3. Draw a circle of radius 5 cm. From a point 12 cm away from its centre, construct the pair of tangents to the circle and measure their lengths.

4. Construct a tangent to a circle of radius 8 cm from a point on the concentric circle of radius 12 cm and measure its length. Also verify the measurement by actual calculation.

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83



Parameter

Able to construct

perpendicular bisector of

a given line segment

Able to construct tangents

to a given circle from an

external point.

Able to find the centre of

a given circle

84


Post Content (PCW1)


1. PT is the tangent to a circle with centre O, OT = 56 cm, TP = 90 cm, find OP.

Which theorem have you used?

2. PT is a tangent to a circle with centre O, PT = 36 cm, AP = 24 cm. Find the radius of

the circle.

3. From a point P, 10 cm away from the centre of the circle, tangent PT of 8 cm is

drawn. Find the radius of the circle.

4. Draw a circle with centre O. Draw diameter AB. Now, draw tangents at the end

points of diameter. Are these parallel or intersecting? Justify your answer.

5. Find the actual sides of the ∆OTP.

85

6. TP and TQ are the two tangents to a circle with centre O so that = 130⁰.

Find

7. From a point Q, the length of tangent to a circle is 40 cm and the distance of Q

from the centre is 41 cm. Find the radius of the circle.

8. The common point of a tangent to a circle with the centre is called ___________ .

9. The length of a tangent from a point A at a distance 5 cm from the centre is 4 cm.

Find the radius of the circle.

86


Post Content (PCW2)


Observe and answer:

1. Observe the following figure and write four observations.

2. Find the perimeter of DEFG. Write justification for the steps used.

87

3. Observe the following figure and write four observations.

4. In the given figure, find x.

88


Post Content (PCW3)


1. What are the instruments to be used in performing constructions?

2. When do you say a line is the perpendicular bisector of another line?

Out of angles of 35o, 40o, 57o and 75o, which can be made with the help of a ruler and

compass?

3. What is a tangent?

4. What can you say about the length of tangents from an external point to a circle?

5. Draw a pair of tangents to a circle of radius 6cm which are inclined to each other at

are angle of 60o. Also justify the construction. Measure the distance between the

centre of the circle and the point of intersection of tangents.

6. Give a method to locate the centre of a circle if it is not given to you.

89


Post Content (Pcw4)


The following word puzzles are statements related to the chapter. Rewrite these

statements so that they become meaningful.

1.

2.

.

90

3.

4.

91

SUGGESTED VIDEOS AND EXTRA READINGS

Definition of tangent


Radius and tangent of a circle


Tangent Segments to a circle (length of tangent)


Inscribed angle

http://bcove.me/ykurgkk5

Angles in semicircles and chords to tangents

http://bcove.me/jxdtwo48

Common Internal and External Tangents


Finding length of tangent when radius is given


Interactive Sheet on tangent to a circle

http://www.mathwarehouse.com/geometry/circle/tangent-to-circle.php#

Construction of tangent at a point on the circle


Finding centre of a circle


Circle through 3 points


Constructing tangents through an external point




http://bcove.me/ykurgkk5

http://bcove.me/jxdtwo48



http://www.mathwarehouse.com/geometry/circle/tangent-to-circle.php




92


Solving questions on circles and tangents

http://www.youtube.com/watch?v=JgF29plLDD8&feature=related

http://www.youtube.com/watch?v=OfIvTh2gqA8&feature=related

http://www.youtube.com/watch?v=uHal7hqIFjw&feature=related

http://www.youtube.com/watch?v=pO0RIUvg1K4&feature=related

http://www.youtube.com/watch?v=tIEAQjrv8Os&feature=related

Properties of tangents:

http://www.youtube.com/watch?v=j0yuSLl8QGc&feature=related

http://www.youtube.com/watch?v=BrNA6G4THUo&feature=related

http://www.youtube.com/watch?v=WejESWLyrps&feature=related

http://www.youtube.com/watch?v=1pPGNu-ZM-0&feature=related


http://www.youtube.com/watch?v=JgF29plLDD8&feature=related

http://www.youtube.com/watch?v=OfIvTh2gqA8&feature=related

http://www.youtube.com/watch?v=uHal7hqIFjw&feature=related

http://www.youtube.com/watch?v=pO0RIUvg1K4&feature=related

http://www.youtube.com/watch?v=tIEAQjrv8Os&feature=related

http://www.youtube.com/watch?v=j0yuSLl8QGc&feature=related

http://www.youtube.com/watch?v=BrNA6G4THUo&feature=related

http://www.youtube.com/watch?v=WejESWLyrps&feature=related

http://www.youtube.com/watch?v=1pPGNu-ZM-0&feature=related


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circles and constructions - nims dubai · ms. seema lakra, s o ms. preeti hans ... circles and...

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