The Geometry of a Circle - By: Dennis Kapatos

The Geometry of a Circle
Geometry (Grades 10 or 11)
A 5 day Unit Plan using Geometers Sketchpad, graphing calculators,
and various manipulatives (string, cardboard circles, Mira’s, etc.).
Dennis Kapatos
I2T2 Project
12/1/05

Unit Overview
Unit Objectives:
Students will learn a broad range of skills and content knowledge. In addition to all the
theorems in each section, students will be able to make observations and conjectures and to test
these conjectures using the technology and manipulatives at their disposal. Students will also be
able to work cooperatively with other group members to investigate the properties of geometric
figures (circles more specifically) and prove theorems. In addition to these, students will be able
to recognize applications of circles and their related parts in the world around them.
NCTM Standards:
Number and Operation
Students judge the reasonableness of numerical computations and their results.
Algebra
Students draw reasonable conclusions about a situation being modeled.
Geometry
Students explore relationships (including congruence and similarity) among
classes of two- and three-dimensional geometric objects, make and test
conjectures about them, and solve problems involving them.
Students establish the validity of geometric conjectures using deduction, prove
theorems, and critique arguments made by others.
Students use Cartesian coordinates and other coordinate systems, such as
navigational, polar, or spherical systems, to analyze geometric situations.
Students use geometric ideas to solve problems in, and gain insights into, other
disciplines and other areas of interest.
Measurement
Students make decisions about units and scales that are appropriate for problem
situations involving measurement.
Problem Solving
Students build new mathematical knowledge through problem solving.
Students solve problems that arise in mathematics and in other contexts.
Reasoning and Proof
Students make and investigate mathematical conjectures.
Students develop and evaluate mathematical arguments and proofs.
Students select and use various types of reasoning and methods of proof.
Communication
Students organize and consolidate their mathematical thinking through
communication.
Connections
Students recognize and apply mathematics in contexts outside of mathematics.
New York State Standards:
G.PS.6 Use a variety of strategies to extend solution methods to other problems.
G.PS.8 Determine information required to solve a problem, choose methods for obtaining
the information, and define parameters for acceptable solutions.
G.CM.5 Communicate logical arguments clearly, showing why a result makes sense and
why the reasoning is valid.

G.CN.7 Recognize and apply mathematical ideas to problem situations that develop
outside of mathematics.
G.R.1 Use physical objects, diagrams, charts, tables, graphs, symbols, equations, or
objects created using technology as representations of mathematical concepts.
G.R.3 Use representation as a tool for exploring and understanding mathematical ideas.
G.G.27 Write a proof arguing from a given hypothesis to a given conclusion.
G.G.29 Identify corresponding parts of congruent triangles.
G.G.49 Investigate, justify, and apply theorems regarding chords of a circle.
G.G.50 Investigate, justify, and apply theorems about tangent lines to a circle.
G.G.51 Investigate, justify, and apply theorems about the arcs determined by the rays of
angles formed by two lines intersecting a circle.
G.G.52 Investigate, justify, and apply theorems about arcs of a circle cut by two parallel
lines.
G.G.53 Investigate, justify, and apply theorems regarding segments intersected by a
circle.
Materials and Equipment:
Resources:
Geometer’s Sketchpad
Computers
Projector
Graphing Calculators
Compasses
Mira’s
Rulers
Protractors
String
Cardboard Circles
Empty Cans
Textbook: “New York Math A/B: An Integrated Approach – Volume 2”
Bass, Hall, Johnson, and Wood. “New York Math A/B: An Integrated Approach –
Volume 2.” Teacher’s Edition. Prentice Hall, 2001. Chapter 12. Pages 584-631.
Bennett, Dan. “Exploring Geometry with The Geometer’s Sketchpad.” 4 th Edition. Key
Curriculum Press, 2002. Chapter 6. Pages 117 – 130.
Unit Description:
This unit is designed to cover chapter 12 sections one through five. Although the chapter
has 6 sections, this remaining section could be covered similar to the previous five. The five
lessons are intended to be inquiry based, though some of the teacher’s instructions may not seem
so. The teacher should give only as much help as is needed to get the students thinking through a
situation. Also, GSP is used throughout the unit; sometimes as a worksheet in which students fill
in answers while looking at problems, sometimes as a tool for experimenting and conjecturing,
and other times for modeling and solving problems. Additionally, most lessons start with a
problem that leads into the lesson so that students see how the need for more sophisticated ways
of thinking arise from real life problems. (Note: Although the theorems are stated on the last page

of this document, the lessons will make more sense if you have a copy of the textbook to look at
in front of you.)
Lesson Summaries:
Lesson 1
In this lesson, though the worksheet is interactive, students will use GSP merely
to type in answers to questions which should lead them into discovering the
general equation for a circle. By doing the work on GSP, students also become
more familiar with it for future activities. It is not as inquiry-based as the other
lesson, but this is because it is developing a somewhat difficult concept.
Lesson 2
This lesson starts with a problem, involving satellites, in which students first gain
a grasp of it though the use of manipulatives and measurement (circles, strings,
protractors, etc.). They then move on to GSP to explore the situation though
inquiry and discovery. The initial questions leads the students thorough all the
theorems they must learn, and at the end, the students apply these theorems to
find the exact answer to the original problem.
Lesson 3
In this lesson, students again start with manipulatives (a Mira, compass, ruler,
and can (for circle tracing)) to explore a problem and then move on to GSP to
discover more theorems about chords and arcs.
Lesson 4
This lesson begins with a problem that is not as clearly related to the real world
as the others. An interest in problem is developed however because of the
surprising results of inscribing a quadrilateral in a circle. Students again switch to
GSP to do more investigating and they are eventually able to understand and
prove the results of the question after discovering some new theorems.
Lesson 5
This lesson’s theorems are shown in a way that they build directly off of the ones
from the previous lesson. Even so, the theorem that the students are trying to
formalize isn’t easy. Students begin the class using GSP this time. The students
eventually use GSP’s very unique graphing capabilities to provide another model
for discovering the relationship between the different parts of the problem.

Lesson 1: Introduction to Circles
Discovering the Equation of a Circle
Name: Dennis Kapatos
Grade: 11
Subject: Geometry
Materials and Handouts:
Teacher’s Computer with GSP
Projector
Student’s Computers with GSP
GSP Worksheet File: “Problem with Watering Crops”
Textbooks
Graphing Calculators
Lesson Objectives:
● Students will be able to write and apply the two general equation of circle, one centered
at the origin and one at any point (h,k) (Theorem 12.1).
● Students will be able to state where the two general equations come from and derive them
using this knowledge.
● Students will be able to graph a circle on their graphing calculators.
Anticipatory Set:
1. Teacher will have review questions on the board including what the distance formula is and
some examples to apply it to.
2. Teacher will go over the answers with the students where the distance formula comes from.
Developmental Activity:
3. Teacher will talk about the farming industry in this country and how more food is needed from
less land using less labor.
4. Students will log onto their computers and open the GSP worksheet file: “Problem with
Watering Crops” and begin working on it by themselves.
5. Students will work until they finish question 10.
6. Teacher will go over and discuss the answers the students came up with.
7. Students will continue working until the finish question 15.
8. Again the teacher will go over and discuss the answer the students came up with paying special
attention to question 14 (it can be confusing).
9. The students will complete the rest of the worksheet.
10. Again the teacher will go over and discuss the students’ answers
11. The students will submit the completed worksheet to the teacher using the computer.
12. The students will complete the reflection questions to be added to their notebooks.
13. Finally, the teacher will show the students how to use their graphing calculators to graph
circles and discuss why a positive and negative form of the same equation are necessary to get
both halves of the circle. Teacher will give an example of a system of equations to show how this
is useful.
Homework:
On page 589 of the textbook, questions 1-18, 21-30, 32, 34, and 42

Problems with Watering Crops
Name: ________________
Date: _________________
Answers in Boxes
in Red
Farmer Bob is using a new high-tech watering system which uses a computer and many
high powered sprinklers to water his crops for him. To organize his land, Bob decides to
divide his entire field into 100 meter square sections. Sprinklers are placed strategically
in order to water as much of the land as possible without overlap (see figure).
1) What do you notice about this arrangement of sprinklers?
Some areas don’t get watered.
The property line for
Bob’s Land is in Red.
The 100 meter square
sections are in blue.
The red dots are the
sprinkler heads with
their spraying area
shown in green.
2) Estimate: About what percent of Bob’s fields do you think are not getting any water?
Responses Vary.

Notice that some areas of Bob’s field get no water at all. The computer uses the center
sprinkler as the origin, that is, it assigns it the coordinates, (0,0). All other points on
Bob’s field, such as the point (100,0), are measured from this point.
(0,0)
(100,0)
Notice that that this
point, (90,60), isn’t
getting any water.
3) A water sensor placed in the ground tells Bob that the coordinates of one of the points
which is not receiving water are (90,60). Fortunately, the sprinklers can be programmed
to spray different distances. What distance should Bob tell the sprinkler at point (0,0) to
spray in order for it to reach this point? Obviously you could set the sprinkler to spray a
much larger area then needed, but his would waste water. What is the exact distance from
the sprinkler to the dry point? (Show all work.)
d = √( (90-0) 2 + (60-0) 2 ) ≈ 108 meters
4) There are other points in this area as well. What distance should Bob tell the sprinkler
to spray in order to water any given point (x,y)?
√( (x) 2 + (y) 2 )
Distance = _____________
5) Bob’s computer refers to distance as the letter r and it doesn’t like square root signs
either. How could you rewrite the equation above so Bob can enter the distance into the
computer?
__________=___________________
r 2 = (x) 2 + (y) 2 r = 5, cntr = (0,0)
The above equation will give the sprinkler a new spraying radius that will reach any point
(x,y) on the coordinate grid. It is the general equation of a circle with its center located at
the point (0,0) and with a radius of r.
6) What is the radius and center of the circle formed by the equation 5 2 = x 2 + y 2 ?
7) What are they for the circle 36 = x 2 + y 2 ?
r = 6, cntr = (0,0)
8) What is the equation of a circle with center (0,0) and radius 10? With radius 4?
100 = x 2 + y 2 16 = x 2 + y 2

9) What is the equation of the circle for the sprinkler at (0,0), before it’s setting was
changed?
100 2 = x 2 + y 2
10) Name 4 points that lay on the edge of this circle.
(100,0), (0,100), (-100,0), (0,-100)
Bob decides that changing the range of all the sprinklers will create too much overlap and
thus waste too much water (water is expensive in this part of the country), so he decides
to install a new sprinkler at the point (100,58).
(0,0)
(100,0)
11) If this new sprinkler can exactly reach point (100,43) then what is its spraying radius?
Use the distance formula and show your work.
r = √( (100-100) 2 + (58-43) 2 ) = 15 meters
12) If this new sprinkler can exactly reach point any point (x,y) then what is its spraying
radius? (Note: This will be an equation in terms of r, x, and y.)
√( (100 - x) 2 + (58- y) 2 )
r = __________________________
r 2 = (100 - x) 2 + (58- y) 2
Now rewrite this without a square root sign: ______________________________
13) Inside the pair of parentheses of this equation should be minus signs. If the x or y
come after the minus sign then switch them with the other number. Write this equation.
__________________________________
r 2 = (x - 100) 2 + (y - 58) 2
14) What is this the same as doing and why is it allowed in this case? That is, why
doesn’t it change the validity of the equation in this case? Convince yourself that this
doesn’t change anything before moving on.
Reversing the order is the same as multiplying everything
inside the parentheses by a -1, which, because the result is
squared afterwards, doesn’t effect the equation.
(a - b) 2 = (-1 (a - b)) 2 = (-a + b) 2 = (b – a) 2

15) Fill in the blank: The equation in #14 is the general equation of a circle with its center
located at the point _________ and with a radius of r
(100,58)
16) Bob wants to install a lot of these new sprinklers in all the other dry areas as well.
What is the equation for a circle that is centered at any given point (h,k) and that can
reach any point (x,y)? (Again make sure that the x and y come first.)
r 2 = (x - h) 2 + (y - k) 2
__________________________________
Now with this form of the equation, you can write the equation of any circle you could
possibly imagine.
17) What is the radius and center of the circle 7 2 = (x-1) 2 + (y-1) 2 ?
18) What are they for the circle 4 = (x+3) 2 + (y-1) 2 ?
19) What is the equation of a circle with center (4,-5) and radius 9?
r = 7, cntr = (1,1)
r = 2, cntr = (-3,1)
81 = (x - 4) 2 + (y + 5) 2
20) What is the equation of a circle with center (-1,0) that passes through the point (3,2)
20 = (x + 1) 2 + y 2

Reviewing Major Ideas: “Problem with Watering Crops”
Because this is a reflection it
is for the students’ use only.
Also responses will vary
_
1) Describe how the distance formula and the equation of a circle centered at the origin
are related? Why does this intuitively make sense?
2) Will farmer Bob ever get this entire field watered without any overlapping areas? Why
or why not?
3) Write the general equations that you found below for future reference.
THM 12.1 General Equation of Circle with Radius r Centered at the Origin:__________________
THM 12.2 General Equation of Circle with Radius r and Center (h,k): ______________________
4) How did you determine the first equation? What did you do?
5 How did you determine the second equation? What did you do?
6) It’s easy to make mistakes when first encountering applying these formulas. What
mistakes did you make in questions 6-8 and 17-20 (such as squaring r, making h
negative, etc)? What things can you think of that will help you remember not to make
them again?

Lesson 2: Tangents
Discovering and Applying Tangents to Real Life Situations
Materials and Handouts:
Teacher’s Computer with GSP
Projector
Student’s Computers with GSP
Textbooks
Calculators
String
Measuring Tape
Rulers
Protractor
Pre-measured Cardboard Circles
Name: Dennis Kapatos
Grade: 11
Subject: Geometry
Lesson Objectives:
● Students will be able to write/describe and prove Theorems 12.2 through 12.4.
● Students will be able to model problem situations using circles, tangents, radii, and other
lines drawn to a circle.
● Given a geometric figure with missing measurements, students will be able to apply
properties of circles, radii, and tangents to find them.
● Students will be able to work cooperatively to find an estimate to a given geometric
problem.
Reviewing Homework:
1. Answers to the homework questions will be shown with the projector. Students will correct
their own homework.
2. Teacher will go over any problems the majority of the students had difficulty with.
Developmental Activity:
3. Teacher will present today’s problem situation involving satellites on white board.
4. Teacher will discuss with the class how they might solve this problem. They will decide to first
do a rough scale model using a cardboard circles and string and take a more sophisticated
approach later.
5. Teacher will split class into groups of 5. There will be a 2 people to hold the ends of the
tangent strings (though they won’t call it that, yet), 1 satellite person to hold the two strings
together at the proper distance, 1 person to measure the central angle, and 1 person to record the
data.
6. Each group will find their own measurement and the class will share their findings afterwards.
7. In order to find a more exact answer, Teacher will ask the class to break up their groups and go
to their computers to draw the situation on GSP.
8. Teacher will talk the students through how to create the drawing using the teacher’s
instructions below. Teacher will ask questions when indicated.

9. Teacher will tell students to enter the 3 new theorems into their notes as they go, they’ll prove
12.4 for homework.
10. After experimenting with GSP, teacher will work with student to solve original question (they
will have to identify a right triangle and use some right angle trigonometry.
11. Teach will have student’s summarize what they’ve learned (the three theorems) and how they
used the theorems to get an exact answer.
Homework:
Read proof of Theorem 12.2 on page 594, prove Theorem 12.4, and on page 596 of the textbook
questions 1-8, 14-16, and 22

Satellites
There are thousands of satellites circling the Earth right
now tens of thousands of miles from the surface. They
are used for any from relaying cell phone calls and
televisions programming, to tracking weather systems
and locating ships using GPS (global positioning
systems). They can only see a portion of the earth’s
surface at a time but the higher a satellite’s altitude, the
more it can see.
How far can the satellite see around the Earth (in
degrees) from an altitude of 22,236 miles? (Note: The
radius of the Earth is 3960 miles.)
The above altitude puts the satellite at a
geosynchronous orbit, how much could it see if it had a higher orbit?
Satellite
Earth
Altitude = 2.72 cm
Visible Area = 5.17 cm
Can a satellite see half-way around the earth, that’s 180 degrees, if it has a high enough
orbit? Why?

Words in () are what the students should be
guided to conjecture. They should not be told
to them.
Teacher’s Instructions
Instructions (say aloud):
Construct a circle AB
Construct line AC through the center of circle AB;
hold down shift to make it vertical
Construct point D on line AC
Construct ray DE and ray DF as shown
Hide points B and C and line AC
Construct segment DA
Construct point G, the intersection of segment DA and
circle A
Swing rays DE and DF till they exactly touch the
circle, like you did with the string
Construct radius AH and AI to the points where it
looks like the rays touch the circle, make sure
H and I are on the rays, not the circle
Questions:
Does this seem like a very accurate way of
constructing this? What do you notice about the
radii and rays? (they almost look perpendicular)
Check this, measure angle AID and angle AHD,
what did people get? Let’s explore this further
Instructions:
Construct a new circle to the right
Construct a radius
Select it and its endpoint and construct a perpendicular line
Questions:
What happens when you move the point around? This line is called a tangent because it
intersects the circle at exactly one point. That point is called the point of tangency. What
should our definition of a tangent be? This is Theorem 12.2 and 12.3. From any one
point outside the circle, what how many different tangents can be drawn to one circle?
Instructions:
Construct a line through the center of the circle like you did for the other circle
Double click it to mark it as a mirror line
Select the point of tangency, the tangent, and the radius and reflect them
Hide the mirror line and it’s point
Construct the intersection of these 2 tangents.
Hide the 2 tangent lines
Construct the tangent segments
Construct the center segment and the point where it intersects the circle as
shown
Move the first point of tangency that you made around
Questions:
What do you notice about the lengths of the two tangent segments? (they’re
always the same length) Check this, measure them. Now move them around.
does this conjecture check? This is Theorem 12.4. Now getting back to our
question, we know all we need to in order to solve this question.
E
B
C
D
A
m ! A H D = 92.06 ° G
m ! A ID = 93.00 °
E
H
D
A
V
I
W
M
F
F
U
O

Lesson 3: Properties of Chords and Arcs
Using Properties of Chords and Arcs to Solve Problems
Materials and Handouts:
Teacher’s Computer with GSP
Projector
Student’s Computers with GSP
GSP Worksheet Files
Textbooks
Graphing Calculators
Compasses
Rulers
Mira’s
Empty Cans
Name: Dennis Kapatos
Grade: 11
Subject: Geometry
Lesson Objectives:
● Students will be able to utilize Theorems 12.8 to construct the center of a circle.
● Students will be able to write/describe and prove Theorems 12.6, 12.5, 12.7, and 12.9.
● Students will be able to model problem situations using circles, tangents, radii, and other
lines drawn to a circle.
● Given a geometric figure with missing measurements, students will be able to apply the
theorems learned so far to find them.
● Students will be able to work cooperatively to make and test conjectures of geometric
figures.
Reviewing Homework:
1. Answers to the homework questions 1-8, 14-16, and 22 will be shown with the projector.
Students will correct their own homework.
2. Teacher will go over any problems the majority of the students had difficulty with.
3. Teacher will ask one student to present their proof of Theorem 12.4 to the class for discussion.
Anticipatory Set:
4. Teacher will have students construct a point, a circle around this point, and then ask them to
construct the circle’s tangent using only a compass and ruler.
5. Teacher will ask students questions to remind them of relationship between tangent and radius
to the point of tangency.
Developmental Activity:
6. Teacher will ask student how they would construct a tangent without being given the center of
the circle, only the edge.
7. Teacher will present today’s problem situation involving a satellite dish.
8. Teach will have students work with a partner, each doing his/her own work but just sharing
thoughts.

9. Teacher will tell students to construct an arc of a circle using the bottom of a can, or anything
else that will serve this purpose. The idea is that they don’t have the hole at the center like they
would from using a compass.
10. Teacher will ask students how they could find the center if they used a Mira. If students need
help, teacher will tell them to construct a point near both ends of the arc.
11. If students need more help, teacher will tell them to use their Mira find a position where it
maps one of these points onto the other.
12. If students still need more help, tell them to draw this line that the Mira is on top of when it
maps the two points onto each other.
13. Teacher will ask the students questions about what this line is and lead them to the idea that
this line must go through the center because it is a line of symmetry for the circle.
14. Students should recognize the need to repeat this to obtain another line and an intersection.
15. Teacher will ask students to connect the two pairs of points they mapped to each other with
segments at tell them that they are called chords.
16. Teacher will ask students to come up with a definition for a chord.
17. Teacher will ask what they constructed to the chord that found the center (a perpendicular
bisector to the chord, be definition of a perpendicular bisector).
18. Teacher will have students make conjecture of this (Theorem 12.8).
19. Teach will have student break groups, go to their computers, and start up GSP
20. Teacher will have student’s open a pre-made sketch of a circle, a chord, and its
perpendicular bisector.
21. Student’s will more around the points to see that any chord’s perpendicular
bisector crosses the center of the circle (see first picture).
22. Teacher will discuss with class and have students formalize their conjecture and
write Theorem 12.8 in their notes.
22. Teacher will have students look on anther page of this sketch of the satellite from
the beginning problem.
23. Teacher will ask students to find the placement of the receiver (see second
picture).
24. Teacher will students look at other pages of the sketches which will have them
discover Theorems 12.5, 12.6, and 12.9. They are very obvious and thus not too
much time is devoted to them. (see third and fourth pictures).
25. Students will enter these theorems into their notes.
26. Students will spend rest of class trying to solve question 20 on page 604.
?
Homework:
Length D'E on
m D'E = 4.83 cm
BC = 5.13 cm
E
drag
a a = 68.76 °
Prove Theorem 12.9, and on page 603 of the textbook, questions 1-13,
18, and finish 20.
G'
D'
B
Length FG' on BC =
m G'G = 4.83 cm
F
m AB = 3.27 cm
G
B
C
A
m CD = 3.3
m GE = 3.09 cm
E
F
D
m EF = 3.06 cm

Locating the Center of the Satellite Dish
A satellite dish in the shape of an arc (a portion of a circle) receives information by
reflecting it off of a dish onto a receiver. During a strong hurricane, a piece of flying
debris broke the receiver off. How could we find the exact place where to put a
replacement receiver in order to receive a signal again?
?

Lesson 4: Inscribed Angles
Exploring Properties of Inscribed Angles and Quadrilaterals Using GSP
Materials and Handouts:
Teacher’s Computer with GSP
Projector
Student’s Computers with GSP
GSP Worksheet Files
Textbooks
Graphing Calculators
Compasses
Rulers
Mira’s
Protractors
Name: Dennis Kapatos
Grade: 11
Subject: Geometry
Lesson Objectives:
● Students will be able to write/describe and prove Theorem 12.10 and its corollaries.
● Students will be able to prove Theorem 12.11, and explain how it can be thought of as a
corollary to Theorem 12.12.
● Given a geometric figure with missing measurements, students will be able to apply the
theorems learned so far to find them.
● Students will be able to work cooperatively to make and test conjectures of geometric
figures.
Reviewing Homework:
1. Answers to the homework questions 1-13, and 18 will be shown with the projector. Students
will correct their own homework.
2. Teacher will haves students go over their solutions to question 20 and any questions that the
majority of students had trouble with.
3. Teacher will ask one student to present their proof of Theorem 12.9 to the class for discussion.
Anticipatory Set:
4. Teacher will ask student to construct any quadrilateral using compasses, rulers, and/or Mira’s.
5. Teacher will ask students to measure all the angles of their quadrilateral.
6. Students will share what their angle measurements are and conclude that any angle can have
any measure.
Developmental Activity:
6. Teacher will ask students to construct a large circle with any quadrilateral in it, with its
vertexes on the circle (an inscribed quadrilateral).
7. Again, students will measure their angles and look for a relationship (opposite angles are
supplementary).

8. Class will discus why they think this is the case.
9. Teacher will ask students to go to their computers and start up GSP.
10. Teacher will ask class to start drawing quadrilateral but stop after
creating just 2 adjacent sides.
12. Teacher will have students construct central angle to this arc, measure
its arc angle, measure the arc angle of the inscribed angle, and conjecture
the relationship between the inscribed and central angles (this is Theorem
12.10, the inscribe should be half the central angle. Teacher will have
students come up with definitions for these terms as he introduces them.
13. Students will move the points around to test this conjecture and enter
Theorem 12.10 into their notes (see first picture).
14. The teacher will ask the class what if a second angle AEC where
drawn or A”any-point”C were drawn (it would have the same arc
angle)(see second picture).
15. Students will experiment with their sketches, discuss what they think,
discover Corollary 1 to Theorem 12.10, and enter this into their notes.
16. Teacher will ask students to hide angle AEC
17. Teacher will ask students what if A and C were opposite each other on
circle, that is, what if they were on opposites sides of a diameter (angle
ABC would always be equal to 90)(see third picture).
18. Students will move points around on their sketches, discuss what they
think, discover Corollary 2 to Theorem 12.10, and enter this into their
notes (see third picture).
19. Teacher will ask students to make angle ADC less than 180 again.
20. Teacher will ask student to see if Theorem12.10 holds Hide as Raypoint B gets
really close to point C (at this point BC will become a tangent to the circle at
point C).
21. Students will try this, discuss what they see and think (see picture).
22. Students will discuss how line BC becomes a tangent, discover Theorem
12.11, see how it is really just another corollary to Theorem 12.10, and enter
it into their notes as well.
23. Teacher will asks students to return to their original question about
inscribed quadrilaterals.
24. Students should be able to do this part completely on their own and with
m! A B C = 4 6 . 2 0 °
B
E
m! A E C = 4 6 . 2 0 °
m ! A B C = 4 1 . 2 2 °
B
m ! A B C = 9 0 . 0 0 °
B
D
m ! A B C = 4 8 .1 5 °
D
D
C
D
A
m! A D C = 9 2 . 3 9 °
C
A
m ! A D C = 8 2 . 4 4 °
C
A
m ! A D C = 1 8 0 . 0 0 °
m! A D C = 9 6 .3 0 °
A
B
C
little difficulty (see final picture).
25. Teacher will have them enter this as into their notes as officially,
Corollary 3 to Theorem 12.10.
26. For the rest of the class, students will prove Theorems 12.10 and 12.11
working in pairs and with the teachers assistance. Students will present their
proofs as time allows.
m ! A B C = 4 8 . 1 5 °
m!
A H C = 1 3 1 . 8 5 °
m AHC = 96.30 °
m ABC = 263.70 °
A
Homework:
B
D
H
Questions 1-12, 15, 16, 24-27, and 28 on page 610 of the textbook. Students
will also be asked to come up with and draw and label 1 new real life
application (not one from the book or class) of anything the have learned
from this unit so far. They should involve as many measurements as
possible or necessary.
C

Lesson 5: Angles Formed by Chords and Secants
Coordinate Graphing with GSP – Using Graphs to Determine Geometric Relationships
Materials and Handouts:
Teacher’s Computer with GSP
Projector
Student’s Computers with GSP
GSP Worksheet Files
Textbooks
Graphing Calculators
Name: Dennis Kapatos
Grade: 11
Subject: Geometry
Lesson Objectives:
● Students will be able to write/describe and prove Theorem 12.12 and 12.13.
● Students will be able to recognize real world applications of circles and the theorems
learned so far.
● Students will be able to use coordinate graphs to determine the relationship between parts
of geometric figure.
● Students will be able to work cooperatively to prove geometric theorems.
● Given a geometric figure with missing measurements, students will be able to apply the
theorems learned so far to find them.
Reviewing Homework:
1. Answers to the homework questions 1-12, 15, 16, 24-27, and 28 will be shown with the
projector. Students will correct their own homework.
2. Teacher will ask each student to share the real life applications they have come up with using
the projector to show their drawings etc..
Anticipatory Set:
3. Students will look at the sketches they made on GSP from lesson and teacher will ask questions
to help them remember what theorems they learned.
Developmental Activity:
4. Students will still be on GSP.
5. Teacher will talk the students through how to create the drawing using the teacher’s
instructions below. Teacher will ask questions when indicated. Theorems will be conjectured and
entered into notes as the discussions progress.
6. At the end of the teacher’s instructions, the students will work in pairs to try to prove Theorem
12.13. Teacher will help them though they shouldn’t have too much trouble (they’ve done harder
proofs before).
7. This proof will lead nicely into Theorem 12.12 which the students will also enter into their
notes.

8. For rest of period, students will practice applying these new theorems to some of the
homework questions.
Homework:
Questions 1-9, and 15-18 on page 617 of the textbook.

Words in () are what the students should be
guided to conjecture. They should not be told
to them.
Teacher’s Instructions
Instructions (say aloud):
Construct a circle AB
Hide point B
Construct Segments CD and ED as shown
Questions:
These are a new type of line we haven’t seen
yet. They’re called secants. How might we define them?
m CE on AB = 60.00 °
Instructions:
Construct the intersections, F and G, of these
secants
Measure arc angles CE and FG
Measure angle D
Move points C and/or E so that the measure of arc angle CE is a round number, say 60
degrees
Questions:
As you move point D onto the circle, how does this situation look familiar? (It is the
inscribed angle theorem from last class.) What about when D is inside the circle? Outside? Lets
explore the relationships between these angles.
Instructions:
Select the measurement of angle D then the arc angle of FG
From the graph menu, choose “plot as (x,y)
Position your axis and change your unit values to make the graph fit nicely as shown
The point that was created by the “plot as (x,y)” above, choose it and from the display
menu, choose trace point
Move point D around outside the circle
C
E
A
m FG on AB = 29.02 °
F
G
D
m! C D E = 1 5 . 4 9 °

80
70
60
50
m CE on AB = 60.00 °
m FG on AB = 18.06 °
40
C
A
F
G
30
20
D
m ! C D E = 2 0 .9 7 ° J
E
10
-80 -60 -40 -20 20 40 60 80
-10
Questions:
What is this plotting? What’s going on? What is the relationship between angle D and the
two arc angles you measured? (Students should work until they come up with the idea that the
measure of angle A is half the difference of the two arc angles. They can get this through asking
questions about the graph, its intercepts, it’s slope, etc.)
Instructions:
Move points C and/or E so that they are both tangents
Move D around outside the circle again
Now move points C and/or E so that one is a secant and one is a tangent
Move D around outside the circle a third time
Questions:
Does it matter weather DF and DG are both secants or tangents or combinations of the
two? (No, the formula holds, it doesn’t matter.) This is all three parts of Theorem 12.13, enter this
into your notes, well prove it next
Instructions:
Hide all the measurements, axis, and gridlines
Construct segments CG and FE as shown
Make sure no lines are touching A
Questions:
(Teacher will have students attempt a proof (they know all
they need to from lesson 4). This proof will also lead very nicely
into Theorem 12.12, which students will enter into their notes.)
E
C
A
F
G
D

Theorems for Chapter 12:
12.1 The standard form of an equation of a circle with center (h,k) and radius r is (xh)
2 + (y-k) 2 = r 2 .
12.2 If a line is tangent to a circle, then it is perpendicular to the radius drawn to the
point of tangency.
12.3 If a line in the same plane as a circle is perpendicular to a radius at its endpoint on
the circle, then the line is tangent to the circle.
12.4 Two segments tangent to a circle from a point outside the circle are congruent.
12.5 In the same circle or in congruent circles, 1 congruent central angles have
congruent arcs and, 2 congruent arcs have congruent central angels.
12.6 In the same circle or in congruent circles, 1 congruent chords have congruent arcs
and, 2 congruent arcs have congruent chords.
12.7 A diameter that is perpendicular to a chord bisects the chord and its arc.
12.8 The perpendicular bisector of a chord contains the center of the circle.
12.9 In the same circle or in congruent circles, 1 chords equidistant form the center are
congruent and, 2 congruent chords are equidistant from the center.
12.10 The measure of an inscribed angle is half the measure of its intercepted arc.
Corollary 1
Corollary 2
Two inscribed angels that intercept the same arc are congruent.
An angle inscribed in a semicircle is a right angle.

Corollary 3
The opposite angels of a quadrilateral inscribed in a circle are
supplementary.
12.11 The measure of an angle formed by a chord and a tangent that intersect on a circle
is half the measure of the intercepted arc.
12.12 The measure of an angle formed by two chords that intersect inside a circle is half
the sum of the measures of the intercepted arcs.
12.13 The measure of an angle formed by two secants, two tangents, or a secant and a
tangent drawn from an point outside the circle is half the difference of the
measures of the intercepted arcs.