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MTH-5109-1
GEOMETRY IV
This course was produced in collaboration with the department of the
Secretary of State of Canada.
Production Supervision : Jean-Paul Groleau
Author: Serge Dugas
Content Revision: Jean-Paul Groleau
Desktop Publishing: L’atelier du Mac inc.
English Version: Ministère de l'Éducation
Services à la communauté anglophone
Direction de la production en langue anglaise
Reprint: 2006
©
Société de formation à distance des commissions scolaires du Québec
All rights for translation and adaptation, in whole or in part, reserved for all countries.
Any reproduction by mechanical or electronic means, including microreproduction, is
forbidden without the written permission of a duly authorized representative of the
Société de formation à distance des commissions scolaires du Québec (SOFAD).
Legal Deposit — 2006
Bibliothèque et Archives nationales du Québec
Bibliothèque et Archives Canada
ISBN 978-2-89493-320-6
Answer Key
MTH-5109-1
Geometry IV
TABLE OF CONTENTS
Introduction to the Program Flowchart .................................................. 0.4
The Program Flowchart ........................................................................... 0.5
How to Use This Guide ............................................................................ 0.6
General Introduction ................................................................................ 0.9
Intermediate and Terminal Objectives of the Module ........................... 0.10
Diagnostic Test on the Prerequisites ...................................................... 0.15
Answer Key for the Diagnostic Test on the Prerequisites ..................... 0.19
Analysis of the Diagnostic Test Results .................................................. 0.21
Information for Distance Education Students ....................................... .0.23
UNITS
1.
2.
3.
4.
Identifying Various Elements in Circles ................................................. 1.1
Relationships Governing the Measure of Length in a Circle ................. 2.1
Relationships Governing Angular Measures in a Circle ........................ 3.1
Solving Real-Life Problems Involving the Relationships
Governing Measures in a Circle .............................................................. 4.1
5. Relationships Governing Measures in a Right Triangles ...................... 5.1
6. Solving Real-Life Problems Involving the Relationships
Governing Measures in a Right Triangle ............................................... 6.1
Final Summary......................................................................................... 7.1
Answer Key for the Final Summary ....................................................... 7.4
Terminal Objectives ................................................................................. 7.5
Self-Evaluation Test................................................................................. 7.7
Answer Key for the Self-Evaluation Test ............................................... 7.19
Analysis of the Self-Evaluation Test Results ......................................... 7.23
Final Evaluation....................................................................................... 7.24
Answer Key for the Exercises .................................................................. 7.25
Glossary .................................................................................................... 7.55
List of Symbols ......................................................................................... 7.58
Bibliography ............................................................................................. 7.59
Review Activities ...................................................................................... 8.1
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Geometry IV
INTRODUCTION TO THE PROGRAM FLOWCHART
Welcome to the World of Mathematics!
This mathematics program has been developed for the adult students of the
Adult Education Services of school boards and distance education. The learning
activities have been designed for individualized learning. If you encounter
difficulties, do not hesitate to consult your teacher or to telephone the resource
person assigned to you. The following flowchart shows where this module fits
into the overall program. It allows you to see how far you have progressed and
how much you still have to do to achieve your vocational goal. There are several
possible paths you can take, depending on your chosen goal.
The first path consists of modules MTH-3003-2 (MTH-314) and MTH-4104-2
(MTH-416), and leads to a Diploma of Vocational Studies (DVS) and certain
Cegep-level programs for the module MTH-4104-2.
The second path consists of modules MTH-4109-1 (MTH-426), MTH-4111-2
(MTH-436) and MTH-5104-1 (MTH-514), and leads to a Secondary School
Diploma (SSD), which allows you to enroll in certain Cegep-level programs that
do not call for a knowledge of advanced mathematics.
The third path consists of modules MTH-5109-1 (MTH-526) and MTH-5111-2
(MTH-536), and leads to Cegep programs that call for a solid knowledge of
mathematics in addition to other abilities.
If this is your first contact with this mathematics program, consult the flowchart
on the next page and then read the section “How to Use This Guide.” Otherwise,
go directly to the section entitled “General Introduction.” Enjoy your work!
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Geometry IV
THE PROGRAM FLOWCHART
CEGEP
MTH-5112-1
MTH-5111-2
MTH-536
MTH-5104-1
MTH-5103-1
Introduction to Vectors
MTH-5109-1
Geometry IV
MTH-5108-1
Trigonometric Functions and Equations
MTH-5107-1
Exponential and Logarithmic Functions
and Equations
Optimization II
MTH-5106-1
Real Functions and Equations
Probability II
MTH-5105-1
Conics
MTH-5102-1
Statistics III
MTH-5101-1
MTH-436
MTH-426
MTH-4110-1
MTH-216
MTH-116
© SOFAD
The Four Operations on
Algebraic Fractions
Sets, Relations and Functions
MTH-4108-1
Quadratic Functions
MTH-4107-1
Straight Lines II
MTH-4106-1
Factoring and Algebraic Functions
MTH-4105-1
Exponents and Radicals
MTH-4103-1
MTH-4102-1
MTH-4101-2
Complement and Synthesis I
MTH-4109-1
MTH-4104-2
MTH-314
Optimization I
MTH-4111-2
Trades
DVS
MTH-416
Complement and Synthesis II
MTH-5110-1
MTH-526
MTH-514
Logic
Statistics II
Trigonometry I
Geometry III
Equations and Inequalities II
MTH-3003-2
Straight Lines I
MTH-3002-2
Geometry II
MTH-3001-2
The Four Operations on Polynomials
MTH-2008-2
Statistics and Probabilities I
MTH-2007-2
Geometry I
MTH-2006-2
Equations and Inequalities I
MTH-1007-2
Decimals and Percent
MTH-1006-2
The Four Operations on Fractions
MTH-1005-2
The Four Operations on Integers
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25 hours
= 1 credit
50 hours
= 2 credits
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MTH-5109-1
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Geometry IV
HOW TO USE THIS GUIDE
Hi! My name is Monica and I have been
asked to tell you about this math module.
What’s your name?
Whether you are
registered at an
adult education
center or at
Formation à
distance, ...
Now, the module you have in your
hand is divided into three
sections. The first section is...
I’m Andy.
... you have probably taken a
placement test which tells you
exactly which module you
should start with.
... the entry activity, which
contains the test on the
prerequisites.
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You’ll see that with this method, math is
a real breeze!
My results on the test
indicate that I should begin
with this module.
By carefully correcting this test using the
corresponding answer key, and recording your results on the analysis sheet ...
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... you can tell if you’re well enough
prepared to do all the activities in the
module.
And if I’m not, if I need a little
review before moving on, what
happens then?
Geometry IV
In that case, before you start the
activities in the module, the results
analysis chart refers you to a review
activity near the end of the module.
I see!
In this way, I can be sure I
have all the prerequisites
for starting.
START
The starting line
shows where the
learning activities
begin.
Exactly! The second section
contains the learning activities. It’s
the main part of the module.
?
The little white question mark indicates the questions
for which answers are given in the text.
The target precedes the
objective to be met.
The memo pad signals a brief reminder of
concepts which you have already studied.
?
Look closely at the box to
the right. It explains the
symbols used to identify the
various activities.
The boldface question mark
indicates practice exercices
which allow you to try out what
you have just learned.
The calculator symbol reminds you that
you will need to use your calculator.
?
The sheaf of wheat indicates a review designed to
reinforce what you have just learned. A row of
sheaves near the end of the module indicates the
final review, which helps you to interrelate all the
learning activities in the module.
FINISH
Lastly, the finish line indicates
that it is time to go on to the self-evaluation
test to verify how well you have understood
the learning activities.
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There are also many fun things
in this module. For example,
when you see the drawing of a
sage, it introduces a “Did you
know that...”
It’s the same for the “math whiz”
pages, which are designed especially for those who love math.
For example. words in boldface italics appear in the
glossary at the end of the
module...
A “Did you know that...”?
Yes, for example, short tidbits
on the history of mathematics
and fun puzzles. They are interesting and relieve tension at
the same time.
Geometry IV
Must I memorize what the sage says?
No, it’s not part of the learning activity. It’s just there to
give you a breather.
They are so stimulating that
even if you don’t have to do
them, you’ll still want to.
And the whole module has
been arranged to make
learning easier.
... statements in boxes are important
points to remember, like definitions, formulas and rules. I’m telling you, the format makes everything much easier.
The third section contains the final review, which interrelates the different
parts of the module.
Great!
There is also a self-evaluation
test and answer key. They tell
you if you’re ready for the final
evaluation.
Thanks, Monica, you’ve been a big
help.
I’m glad! Now,
I’ve got to run.
See you!
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Later ...
This is great! I never thought that I would
like mathematics as much as this!
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Geometry IV
GENERAL INTRODUCTION
RELATIONSHIPS GOVERNING THE MEASURE OF LENGTH
IN CIRCLES AND RIGHT TRIANGLES
You are no doubt familiar with words like "circle," "radius," "arc," "area,"
"circumference," "right triangle," "altitude" and "hypotenuse." In all likelihood,
you have seen these concepts in previous courses, without studying them in
depth. In this course, you will expand your knowledge of these different topics,
discover the deductive methods of geometry and learn how to use theorems and
corollaries to justify the steps involved in solving problems.
We will begin by studying the various relationships that exist in a circle. After
briefly reviewing the elements of a circle, we will examine the relationships
governing the measure of length, angles and arcs in a circle.
We will then move on to the relationships governing the measure of length in
right triangles. You are already familiar with one of these relationships: the
Pythagorean theorem. We will discover many others and see the practical
applications of these relationships.
It should be stressed that in all the activities in this guide, the emphasis is on
applying the concepts you learn to solving real-life problems based on concrete
situations.
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Geometry IV
INTERMEDIATE AND TERMINAL OBJECTIVES
OF THE MODULE
Module MTH-5109-1 contains six objectives and requires 25 hours of study,
distributed as shown below. The terminal objectives appear in boldface.
*
Objectives
Number of Hours *
% (evaluation)
1 to 4
12
50%
5 and 6
12
50%
One hour are allotted for the final evaluation.
1. Identifying Various Elements in Circles
On diagrams of circles in which several elements are represented and
labelled with upper case letters, identify the following: a radius, a diameter,
a chord, an arc, a secant, a tangent, a point of tangency, a central angle, an
inscribed angle, an interior angle and an exterior angle.
2. Relationships Governing the Measure of Length in a Circle
Given a list of theorems as well as a diagram of one or two circles with the
measures required to solve a particular problem, determine the measure of
a radius, a diameter, a circumference, an area, a chord, an arc or a tangent
segment, indicating the theorem(s) used to support each step in the solution.
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Geometry IV
The following is a list of the given theorems and corollaries.
Relationships Within a Circle
• Any perpendicular bisector of a chord of a circle is a diameter of a circle.
• The longest chord of a circle is a diameter.
• In a circle, any radius perpendicular to a chord divides that chord into two
congruent segments.
• In a circle, any radius perpendicular to a chord divides the subtended arc
into two congruent arcs.
• In a circle, arcs located between two parallel chords are congruent.
• Two chords are congruent if they are equidistant from the centre of the
circle.
• In a circle, congruent chords subtend congruent arcs and, conversely,
congruent arcs are subtended by congruent chords.
• Any line tangent to a circle is perpendicular to the radius that shares the
point of tangency.
• For any circle, two tangent segments originating from the same exterior
point are congruent. (The segments are measured from that exterior point
to their respective points of tangency).
• Two parallel lines, be they tangents or secants, intercept congruent arcs
of a circle.
Relationships Involving Two Circles
• The circumferences of two circles have the same ratio as their radii.
• The areas of two circles have the same ratio as the squares of their radii.
• The measures of similar arcs of two circles have the same ratio as their
radii.
Each solution should involve the application of no more than three theorems
or corollaries.
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Geometry IV
3. Relationships Governing Angular Measures in a Circle
Given a list of theorems and corollaries as well as a diagram of a circle with
the measures required to solve a particular problem, determine the measure
of a central angle, an inscribed angle, an interior angle, an exterior angle or
the measure of an arc in degrees, indicating the theorems or corollaries used
to support each step in the solution.
The following is a list of the given theorems and corollaries.
• In a circle, the measure of a central angle is equal to the measure of its
intercepted arc.
• In a circle, the measure of an inscribed angle is one-half the measure of its
intercepted arc.
• In a circle, the measure of an interior angle is equal to one-half the sum
of the measures of the arcs intercepted by the angle and its vertical angle.
• For any circle, the measure of an exterior angle is equal to one-half the
difference of the measures of the intercepted arcs.
Each solution should involve the application of no more than three theorems
or corollaries.
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Geometry IV
4. Solving Real-Life Problems Involving the Relationships Governing
Measures in a Circle
Given diagrams of one or two circles with the dimensions and
angular measures that make it possible to draw relevant conclusions
as well as a list of theorems and corollaries pertaining to the relationships governing measures in a circle, solve problems related to a
variety of human activities (e.g. carpentry, land surveying, architecture and technical drawing). The measure of the required angle,
radius, diameter, segment, perimeter or area must be stated in the
appropriate unit. The steps in the solution must be shown and the
theorems and corollaries used to support the answer must be indicated.
5. Relationships Governing Measures in a Right Triangle
Given a list of theorems and corollaries as well as a diagram of a right triangle
with the measures required to solve a particular problem, determine the
measure of an angle, a side, a median, an altitude, the hypotenuse, the
perimeter or the area of a right triangle, indicating the theorems or corollaries used to support each step in the solution.
The following is a list of the given theorems and corollaries.
• The hypotenuse of a right triangle inscribed in a circle is always a
diameter of that circle.
• In a right triangle, the length of the median to the hypotenuse is one-half
the length of the hypotenuse.
• In a right triangle the length of the side opposite a 30° angle is one-half
the length of the hypotenuse.
• A right triangle and the altitude to its hypotenuse form two right triangles
that are similar to the given triangle and to each other.
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Geometry IV
• The length of the altitude to the hypotenuse of a right triangle is the
geometric mean between the lengths of the segments of the hypotenuse.
• The length of a leg of a right triangle is the geometric mean between the
measure of its projection onto the hypotenuse and the measure of the
hypotenuse.
• The product of the lengths of the legs of a right triangle is equal to the
product of the length of the hypotenuse and the length of the altitude to
the hypotenuse.
Each solution should involve the application of no more than three theorems.
6. Solving Real-Life Problems Involving the Relationships Governing
Measures in a Right Triangle
Given the diagram of a right triangle with the dimensions and
angular measures that make it possible to draw relevant conclusions
as well as a list of theorems and corollaries pertaining to the relationships governing measures in a right triangle, solve problems related
to a variety of human activities (e.g. carpentry, land surveying,
architecture and technical drawing). The measure of the required
angle, side, perimeter or area must be stated in the appropriate unit.
The steps in the solution must be shown and the theorems and
corollaries used to support the answer must be indicated.
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Geometry IV
DIAGNOSTIC TEST ON THE PREREQUISITES
Instructions
1. Answer as many questions as you can.
2. To answer these questions, you must have the following instruments:
• a ruler marked off in centimetres and millimetres,
• a protractor,
• a compass,
• a calculator.
3. Write your answers on the test paper.
4. Do not waste any time. If you cannot answer a question, go on to the
next one immediately.
5. When you have answered as many questions as you can, correct
your answers using the answer key which follows the diagnostic
test.
6. To be considered correct, answers must be identical to those in the
key. In addition, the various steps in your answer should be
equivalent to those shown in the solution
7. Transcribe your results onto the chart which follows the answer
key. This chart gives an analysis of the diagnostic test results.
8. Do only the review activities that apply to each of your incorrect
answers.
9. If all your answers are correct, you may begin working on this
module.
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Geometry IV
1. a) Draw a circle with a 2-cm radius.
b) Using a protractor and a compass, draw a 60° arc starting at point A below.
N.B. The construction should be accurate to within ± 2°.
•
A
2. Arthur wants to use an aerosol spray to waterproof the canvas that covers his
circular swimming pool. One can of spray will cover 20 m2. If the radius of
Arthur's swimming pool is 3 m, will one can of spray do the job?
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Geometry IV
3. Ann wants to fence in her circular garden. If the diameter of the garden is 15
m, what will the length of the fence be?
N.B. Give your answer to the nearest centimetre.
4. Melissa always takes a short cut through a field when she comes home from
school.
Calculate
the
distance
(AC)
that
she
walks
if
mAB = 360 m and mBC = 480 m.
A
B
C
5. Complete the sentence below using one of the following words: altitude,
perpendicular bisector, median.
a) In the figure below, line segment AE is .....................................................
A
B
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Geometry IV
b) In the figure below, we know that mAE = mEC. Hence, line segment BE
is ..................................................
B
A
E
C
c) In the figure below, segment DE is perpendicular to the midpoint of side
AC. Hence, line segment DE is
B
D
A
E
C
6. Find the value of x in the following proportions. Show all the steps in the
solution.
a)
3 = x
13 78
b) 0.2 = 1.4
x
6
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Geometry IV
ANSWER KEY FOR THE DIAGNOSTIC TEST
ON THE PREREQUISITES
1. a)
b)
60°
•
60°
2 cm
A•
2. Determine the area of the swimming pool canvas to be waterproofed.
A = πr2
A = π × (3 m)2 = π × 9 m2 = 28.27 m2 (to the nearest hundredth)
Hence, Arthur will need more than one can of spray to do this job.
3. Find the circumference of the garden to be fenced in.
C = πd
C = π × 15 m = 47.12 m (to the nearest hundredth)
Hence, the length of the fence will be 47.12 m.
4. Use the Pythagorean theorem.
mAC2 = mAB2 + mBC2
mAC2 = (360 m)2 + (480 m)2
mAC2 = 129 600 m2 + 230 400 m2
mAC2 = 360 000 m2
mAC = 600 m
Hence, Melissa walks 600 m.
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MTH-5109-1
5. a) An altitude
6. a)
3 = x
13
78
13 × x = 3 × 78
3
b) A median
Geometry IV
c) A perpendicular bisector
b)
0.2 = 1.4
x
6
0.2 × x = 6 × 1.4
13x = 234
0.2x = 8.4
x = 234
13
x = 18
x = 8.4
0.2
x = 42
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Geometry IV
ANALYSIS OF THE DIAGNOSTIC TEST RESULTS
Answers
Questions Correct
Incorrect
1. a)
b)
2.
3.
4.
5. a)
b)
c)
6. a)
b)
Review
Section
Page
8.1
8.2
8.3
8.4
8.5
8.6.1
8.6.2
8.6.3
8.7
8.7
8.4
8.7
8.12
8.15
8.18
8.24
8.28
8.30
8.33
8.33
Before Going On To
Unit 1
Unit 1
Unit 2
Unit 2
Unit 3
Unit 5
Unit 5
Unit 2
Units 2 and 5
Units 2 and 5
• If all your answers are correct, you may begin working on this module.
• For each incorrect answer, find the related section listed in the Review
column. Do the review activities for that section before beginning the unit
listed in the right-hand column under the heading Before Going On To.
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INFORMATION
FOR
EDUCATION STUDENTS
Geometry IV
DISTANCE
You now have the learning material for MTH-5109-1 together with the homework assignments. Enclosed with this material is a letter of introduction from
your tutor indicating the various ways in which you can communicate with him
or her (e.g. by letter, telephone) as well as the times when he or she is available.
Your tutor will correct your work and help you with your studies. Do not hesitate
to make use of his or her services if you have any questions.
DEVELOPING EFFECTIVE STUDY HABITS
Distance education is a process which offers considerable flexibility, but which
also requires active involvement on your part. It demands regular study and
sustained effort. Efficient study habits will simplify your task. To ensure
effective and continuous progress in your studies, it is strongly recommended
that you:
• draw up a study timetable that takes your working habits into account and
is compatible with your leisure time and other activities;
• develop a habit of regular and concentrated study.
The following guidelines concerning the theory, examples, exercises and assignments are designed to help you succeed in this mathematics course.
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Geometry IV
Theory
To make sure you thoroughly grasp the theoretical concepts:
1. Read the lesson carefully and underline the important points.
2. Memorize the definitions, formulas and procedures used to solve a given
problem, since this will make the lesson much easier to understand.
3. At the end of an assignment, make a note of any points that you do not
understand. Your tutor will then be able to give you pertinent explanations.
4. Try to continue studying even if you run into a particular problem. However,
if a major difficulty hinders your learning, ask for explanations before
sending in your assignment.
Contact your tutor, using the procedure
outlined in his or her letter of introduction.
Examples
The examples given throughout the course are an application of the theory you
are studying. They illustrate the steps involved in doing the exercises. Carefully
study the solutions given in the examples and redo them yourself before starting
the exercises.
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Geometry IV
Exercises
The exercises in each unit are generally modelled on the examples provided.
Here are a few suggestions to help you complete these exercises.
1. Write up your solutions, using the examples in the unit as models. It is
important not to refer to the answer key found on the coloured pages at the
end of the module until you have completed the exercises.
2. Compare your solutions with those in the answer key only after having done
all the exercises. Careful! Examine the steps in your solution carefully even
if your answers are correct.
3. If you find a mistake in your answer or your solution, review the concepts that
you did not understand, as well as the pertinent examples. Then, redo the
exercise.
4. Make sure you have successfully completed all the exercises in a unit before
moving on to the next one.
Homework Assignments
Module MTH-5109-1 contains three assignments.
The first page of each
assignment indicates the units to which the questions refer. The assignments
are designed to evaluate how well you have understood the material studied.
They also provide a means of communicating with your tutor.
When you have understood the material and have successfully done the pertinent exercises, do the corresponding assignment immediately. Here are a few
suggestions.
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Geometry IV
1. Do a rough draft first and then, if necessary, revise your solutions before
submitting a clean copy of your answer.
2. Copy out your final answers or solutions in the blank spaces of the document
to be sent to your tutor. It is preferable to use a pencil.
3. Include a clear and detailed solution with the answer if the problem involves
several steps.
4. Mail only one homework assignment at a time. After correcting the assignment, your tutor will return it to you.
In the section “Student’s Questions”, write any questions which you may wish to
have answered by your tutor. He or she will give you advice and guide you in your
studies, if necessary.
In this course
Homework Assignment 1 is based on units 1 to 4.
Homework Assignment 2 is based on units 5 and 6.
Homework Assignment 3 is based on units 1 to 6.
CERTIFICATION
When you have completed all the work, and provided you have maintained an
average of at least 60%, you will be eligible to write the examination for this
course.
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Geometry IV
START
UNIT 1
IDENTIFYING VARIOUS ELEMENTS
IN CIRCLES
1.1
SETTING THE CONTEXT
The Dart Board
Using a piece of plywood that he found in his garage, Bruno, a talented
handyman, wants to make a dart board for his children, Kathy and Freddy. He
shows them his sketch of the dart board and explains that players will be
awarded points according to the sections of the board where their darts land.
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Fig. 1.1
Geometry IV
Sketch of Bruno's Dart Board
"The game is easy to understand," he says. "If your dart lands between two radii,
you get 10 points; but if it lands between a chord and an arc, you score only
5 points. However, . . ." Kathy interrupts him. "What are radii, Dad?" she asks.
"And what's an arc?" adds Freddy.
Bruno soon realizes that his overly technical vocabulary will just confuse his
children and lead to needless arguments between them.
"I've got an idea," he says. "You can help me build the dart board and while we're
at it, I'll explain the terms used to describe the elements of a circle."
To achieve the objective of this unit, you must be able to identify the
following elements of a circle: a radius, a diameter, a chord, an arc, a
secant, a tangent, a point of tangency, a central angle, an inscribed
angle, an interior angle and an exterior angle.
1.2
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Geometry IV
A circle is the set of all points equidistant from another point
called the "centre" of the circle.
B D•
Figure 1.2 shows a circle with centre O.
Point O is not part of the circle: it is a
point located inside the circle. Points
E•
•
A
O
A, B and C are points on the circle,
C
whereas points D and E are located
outside the circle.
Fig. 1.2
?
Circle with Centre O
Where are the points on line segment OA located in relation to the circle?
...........................................................................................................................
All the points on OA are located inside the circle, except for point A, which is on
the circle.
In Figure 1.2, point O is the centre of the circle and OA, OB and OC are radii of
the circle.
The radius of a circle is a line segment connecting the centre
of the circle with a point on the circle. It is denoted by r.
Since there is an infinite number of points on a circle, we can conclude that there
is an infinite number of radii and that they are of equal length.
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Geometry IV
In Figure 1.3, BD, AC and EF are called chords.
B
A
•
O
E
C
F
Fig. 1.3
D
Chords of a Circle
A chord of a circle is a line segment that connects any two
points of a circle.
Chord AC, which passes through the centre of the circle, is called the
diameter.
A diameter is a chord that passes through the centre of the
circle. It is denoted by d.
?
Is the radius of a circle a chord? Explain your answer.
...........................................................................................................................
...........................................................................................................................
The radius of a circle is definitely not a chord because only one of its endpoints
is on the circle.
1.4
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Geometry IV
In Figure 1.4 below, points A and B are located on the circle with centre O. If we
join all the points between A and B, we obtain minor arc AB. If we wish to
identify major arc AB, we must use a third letter such as C to avoid confusion.
(
This is symbolized by AB and BCA, which means arc AB and arc BCA.
A
Minor Arc AB
•
Major Arc BCA
B
C
Fig. 1.4 Arcs of a Circle
An arc of a circle is a part of the circle contained between any
two points of the circle.
A chord subtends an arc of a circle and an arc of a circle is subtended by a chord.
B
C
D
A
Fig. 1.5 Arcs Subtended by Chords
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Geometry IV
(
(
In Figure 1.5, chord AB subtends AB and chord CD subtends CD. We can also
say that arcs AB and CD are subtended by chords AB and CD respectively.
There are also lines that are neither radii, nor diameters, nor chords. Some are
called secants. Others are called tangents.
?
In Figure 1.6 below, what difference do you notice between lines AB, BC and
QP, and lines XY, YZ and QR?
...........................................................................................................................
...........................................................................................................................
Y
B
R
P
X
A
C
Z
Q
Fig. 1.6 Circles and Lines
Lines AB, BC and QP intersect the circles at two points, whereas lines XY, YZ
and QR touch the circles at only one point. The first three lines are called
secants.
Lines XY, YZ and QR are tangents because they touch the circle at only one
point. This point is called the point of tangency.
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Geometry IV
A secant is a line that intersects a circle at two points.
A tangent is a line that touches a circle at only one point. This
point is called the point of tangency.
F
G
?
H
Name two secants, two tangents and
I
two points of tangency in the adjoin-
E
ing figure.
D
B
A
C
Fig. 1.7 Secants and Tangents
Secants: .................
Tangents: ..............
Points of tangency: ....................
Lines AF and DH are the two secants, whereas lines AD and CF are the two
tangents. Points B and E are the points of tangency.
Now, let's apply these concepts.
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Geometry IV
Exercice 1.1
1. Study the following figure and answer the questions below.
C
B
D
E
A
O
F
G
a) Name five radii. ..........................................................................................
b) Name two diameters. ..................................................................................
c) Name four chords. .......................................................................................
d) Name the two longest chords in the figure. ...............................................
e) What point do both these long chords pass through? ...............................
f) What do we call these two long chords? ....................................................
g) Using three letters, name five different arcs that start with the
letter A. ........................................................................................................
h) Name the chord that subtends BAG. .........................................................
1.8
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Geometry IV
2. Refer to the following figure and answer the questions below.
A
B
J
G
K
O
E
H
F
C
D
Name:
a) all the points located on the circle. ............................................................
b) all the points located inside the circle. ......................................................
c) all the points located outside the circle. ....................................................
d) all the secants. ............................................................................................
e) all the tangents. ..........................................................................................
f) all the radii. .................................................................................................
g) all the diameters. ........................................................................................
h) all the chords. ..............................................................................................
i) all the points of tangency. ..........................................................................
Kathy and Freddy are already able to pick out radii, diameters, secants and
tangents on their dart board. They are no longer going around in circles! To make
things easier to understand, Bruno added a few letters to the dart board.
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Geometry IV
G
C
B
D
H
F
A
Fig. 1.8
O
E
Selected Points on Bruno's Dart Board
"Now look at the angles," says Bruno. "Notice that the vertices of several angles
are inside the circle and even at its centre, some are on the circle and there is one
outside the circle."
An angle is a figure formed by two rays having the same
endpoint. This point is called the vertex of the angle.
A
Angle AOB (denoted by ∠AOB) is
•
formed by the two rays OA and OB.
The vertex of this angle is point O.
O•
•
B
Fig. 1.9 Angle AOB
Let's identify the various angles related to a circle.
1.10
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Geometry IV
An angle whose vertex is located inside a circle is called an
interior angle.
An angle whose vertex is located at the center of a circle is
called a central angle.
O
O
Fig. 1.10 Interior Angles and Central Angles
In Figure 1.10, the two circles on the left contain interior angles. The circles on
the right also contain interior angles, but these are called central angles because
their vertices are at the centre of the circle (point O).
?
In Figure 1.8, name three central angles and three interior angles.
Central angles:
............................................................................................
Interior angles:
............................................................................................
Point O is the vertex of each central angle (e.g. angles COA, DOE, BOD).
Any angle whose vertex is point H, point F or point O is an interior angle (e.g.
angles BHA, CHO, DFE, EFO, AOB).
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Geometry IV
An angle whose vertex is outside the circle and which intersects
the circle in at least two points is called an exterior angle.
1
2
Fig. 1.11
3
Exterior Angles
Note that exterior angles are always formed by secants or tangents.
?
In Figure 1.11, what types of lines form:
• angle 1?
....................................................................................................
• angle 2?
....................................................................................................
• angle 3?
....................................................................................................
Angle 1 is formed by two secants, angle 2 by two tangents and angle 3 by a secant
and a tangent.
?
Name the only exterior angle on Bruno's dart board in Figure 1.8...............
The only exterior angle is ∠BGD.
In some cases, the vertex of an angle is located on the circle; this type of angle is
called an inscribed angle.
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Geometry IV
An inscribed angle is an angle whose vertex is located on the
circle. This type of angle is formed by two chords or by a chord
and a tangent to the circle
Fig. 1.12 Inscribed Angles
In Figure 1.8, there are several inscribed angles (e.g. any angle whose vertex is
A, B, C, D or E is an inscribed angle).
?
In the following figure, state whether the angles are interior, central, exterior
or inscribed.
F
A
G
O
E
B
D
C
Fig. 1.13 Different Angles Associated with the Circle
a) ∠AOD .....................................
b) ∠AED ....................................
c) ∠CDO .....................................
d) ∠OAE ....................................
e) ∠AGF
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Geometry IV
Now, let's check your answers and understanding of what we've done so far.
a) ∠AOD is both an interior angle and a central angle because its vertex is
at the centre of the circle.
b) ∠AED is an exterior angle. Its vertex is outside the circle and the angle
is formed by a secant and a tangent.
c) ∠CDO is an inscribed angle because its vertex is on the circle.
d) ∠OAE is an inscribed angle. Its vertex is on the circle.
e) ∠AGF is an interior angle. Its vertex is inside the circle.
Exercice 1.2
In the figure below, name:
A
B
D
O
C
G
F
E
a) all the central angles.
................................................................................
b) all the interior angles.
................................................................................
c) all the inscribed angles.
................................................................................
d) all the exterior angles.
................................................................................
After all those definitions, don't you think we deserve a break?
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Geometry IV
Did you know that…
..., strange as it may seem, you can calculate the value of π
by dropping a needle on the floor? However, to do this, you
need floorboards of equal width.
It is preferable to use a needle, but a toothpick or any needle-like object
will do. Just make sure that the length of the object is equal to the width
of the floorboards. Then drop the needle repeatedly, recording the number
of times you let it fall and the number of times it lands on a groove between
the floorboards.
Next, double the number of times you have dropped the needle and divide
that figure by the number of times the needle has landed on a groove. The
result will be equal to the value of π. For example, if you drop the needle
100 times and it lands on a groove 62 times, divide 200 by 62. The result
is about 3.2. This is not the exact value of π, but the more you drop the
needle, the closer you will come to the exact value of π!
What accounts for this? When you drop the needle, whether it lands on
a groove depends on the spot where its centre falls and the way in which
the needle turns about its centre. When the needle turns around its
centre, it describes a circle and this is why π, which is directly related to
the measure of the circumference of a circle, is also related to the
probability that the needle will land on a groove.
The mathematician Buffon studied this problem in the eighteenth century and conducted a number of experiments on this subject.
If you're sceptical, try it out with your own needle.
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Geometry IV
PRACTICE EXERCISES
1. Identify the following:
a) The longest chord in a circle: ......................................................................
b) A line that touches the circle at only one point: ........................................
c) An angle formed by two tangents: .............................................................
d) A line segment whose two endpoints are on the circle:
...................................................................................
e) It is not a segment, but its two endpoints are on the circle:
...................................................................................
2. Study the figure below and complete the statements on the following page.
A
E
O
D
B
C
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a) Segment OE is called ......................................................
b) Line AB is called .............................................................
c) Line AC is called .............................................................
d) Point B is called ..............................................................
e) Angle DBC is an ................................................... angle.
f) Angle BAC is an ................................................... angle.
g) Angle DOA is a ..................................................... angle.
h) Segment BC is .................................................................
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Geometry IV
REVIEW EXERCISES
1. To make their dart board, our friends have set themselves up in the attic.
Bruno takes a piece of string, attaching a nail at one end and a pencil at the
other end to draw the circle. "Let me do it, Dad!" exclaims Kathy.
B
B
•O
A
•
A
O
C
What element of the circle do you associate with:
a) the nail? .......................................................................................................
b) the line made by the pencil? .......................................................................
c) the string? ...................................................................................................
d) points A and C? ...........................................................................................
e) the points on the circle between B and C? .................................................
2. After Kathy draws the circle and Bruno cuts it out, Freddy draws the lines to
mark off the different sections for scoring purposes. Then Bruno hangs the
dart board on the garage door. "If everyone agrees," he explains, "here's how
our scoring system will work."
• If a dart lands in the centre: 25 points.
• If a dart lands in a triangle formed by an inscribed angle, a central angle
or an interior angle: 15 points.
• If a dart lands between a chord (other than the diameter) and an arc:
10points.
• If a dart lands on a point of tangency: 3 points.
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Geometry IV
• If a dart lands outside the circle but between two tangents: 2 points.
On the sketch below, write in Bruno's scoring system as accurately as
possible.
I
C
B
D
E
A
O
F
H
G
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Geometry IV
1.4 THE MATH WHIZ PAGE
The Multiplication of Regions
Into how many regions can we divide a circle if we draw one chord, two
chords, three chords, four chords, etc.?
There are two solutions to the problem: one when all the chords are
diameters and another when none of the chords are diameters.
The solution is easy when all the chords are diameters. We can begin
to picture this solution by studying the figure below.
1
1
4
2
2
3
6
1
5
2
3
4
1
8
7
2
6
3
4
5
Fig. 1.14 Regions Created by Diameters
Figure 1.14 shows that each time we add a diameter, the number of
regions increases by two.
?
If we add a fifth and a sixth circle with one more diameter each
time, how many regions will we have?
..........................................................................................
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Geometry IV
We create 10 regions with 5 diameters and 12 regions with 6 diameters.
?
And now if we draw 15 diameters, how many regions will we have?
Try to find the answer without drawing diameters.
.........................................................................................................
If you obtained 30 regions without drawing the diameters, your
answer is right. If you drew them, you still have a problem: counting
the number of regions!
?
What is the mathematical formula that enables us to find the
answer no matter how many diameters have been drawn?
.........................................................................................................
The formula is simply x = 2n where x is the number of regions and n,
the number of diameters.
If, for instance, we have
100 diameters (n), we will have 2(100) regions (x) or 200 regions.
In the next unit we will study the second part of the solution to this
problem.
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