Geometry 16 Day Unit 42 Minute Classes Mr. Nathaniel Frye Beginning Date: March … · 2018. 10. 4. · -Theorem 10.1 (Line is tangent to a circle if and only if the line is perpendicular

Chapter 10: Properties of Circles

Geometry

16 Day Unit

42 Minute Classes

Mr. Nathaniel Frye

Beginning Date: March 6, 2012

The unit on circles and their properties is extremely important in geometry classes

because circles have been used and studied since the beginnings of Geometry. In Euclid's

original five postulates, on which all of Euclidean Geometry is based, it was stated in postulate

four that a circle could be constructed. Through the developments of Geometry, more has

become known about circles and their applications. They are seen everywhere in the world and

have many real life applications which are crucial to understand.

In this unit, students will explore many properties of circles, which include arcs, arc

length, chords, tangents, secants, and many other pieces. There are more than a dozen theorems,

corollaries, and postulates that the students will learn and be able to apply to specific problems

and exercises. Students will also have the opportunity to apply what they are learning to

specially designed performance tasks that focus on real world applications. In constructing their

own knowledge, the students will play a part in discovering some of the theorems by

experimentation and discovery. They will explore connections that exist between the many

theorems and properties of circles. Students will be assessed both informally and formally;

quizzes, performance tasks, exit slips, student feedback, review, and a unit exam will serve as

forms of assessment. After many of these forms of assessment, students will have the

opportunity for revision where further progress can be made.

To meet developmental needs of students, the unit implores the use of many different

literary strategies. Each new lesson greets the students with a highly informative set of notes

with blanks and exercises which the entire class systematically works through. As the content

gets more extensive and difficult, careful scaffolding of exercises and problems is used so

students learn proper techniques of solving problems. To differentiate among different learners,

all of the work is shown on a Promethean Smart-Board and is available online through the school

district website. Furthermore, group work is highly encouraged where students have the

opportunity to share thoughts and ideas with other students.

In meeting Pennsylvania Standards, the unit is planned so the standards are directly met.

In designing the unit, beginning with the standards gives the unit a track that is directly followed

so the goals are properly met. Moreover, the students will learn an extensive amount that will

properly prepare them for higher levels of mathematics, state standardized testing, and specific

standardized tests such as SATs and ACTs.

This lesson on circles is vital for students not only because they will be tested on it, but

also because they will need to grasp the knowledge to perform well in certain professions.

Professions such as carpentry, engineering, and many others utilize circles and their properties

each day. Learning and understanding this knowledge at an earlier age greater prepares these

students for a lifetime of achievement.

Topic: Chapter 10: Properties of Circles

Stage 1: Identify Desired Results

Established Goals/ Standards:

2.9.11.A: Create justifications for arguments related to geometric relations.

-Focuses on:

M11.C.1.1.1: Identify and/or use the properties of radius, diameter and/or tangent

of a circle.

M11.C.1.1.2: Identify and/or use the properties of arcs, semicircles, inscribed

and/or central angles.

M11.C.1.3.1: Identify and/or use properties of congruent and similar polygons

or solids.

2.9.11.B: Use arguments based on transformations to establish congruence or similarity

of 2-dimensional shapes.

-Focuses on:

M11.C.1.1.1: See above



Essential Questions:

Why are circles and the properties of circles important to understand, and how is this

knowledge important in real life?

How can the properties of circles be applied to 3 dimensions?

Enduring Understandings:

Students will understand that...

- at any point on the boundary of a circle, the radius from the center of the circle to that

point is perpendicular to a unique tangent line

- arc lengths are directly related to angle measures

- chord length and angle measures can be derived using congruence and given

lengths/measures of other chords and angles

- inscribed angles are related to central angles

- the equation of a circle depends on the center and radius

Knowledge

Students will know...

- the definitions of the properties of circles (circle, center, radius, chord, diameter, secant,

tangent, central angle, minor/major arc, semicircle, inscribed angle, intercepted

arc, segment of chord, point of tangency, tangent and concentric circles)

- theorems and postulates involving the properties of circles

-Theorem 10.1 (Line is tangent to a circle if and only if the line is perpendicular

to a radius at its endpoint on the circle)

- Theorem 10.2 (Tangent segments from a common external point are congruent)

- Postulate 23: Arc Addition Postulate

- Theorem 10.3 (In the same circle, or in congruent circles, two minor arcs are

congruent if and only if their corresponding chords are congruent)

- Theorem 10.4 (If one chord is a perpendicular bisector of another chords, then

the first chord is a diameter)

- Theorem 10.5 (If a diameter of a circle is perpendicular to a chord, then the

diameter bisects the chord and its arc)

- Theorem 10.7 (The measure of an inscribed angle is one half the measure of its

intercepted angle)

- Theorem 10.8 (If two inscribed angles of a circle intercept the same arc, then the

angles are congruent)

- Theorem 10.9 (If a right triangle is inscribed in a circle, then the hypotenuse is a

diameter of the circle. Conversely, if one side of an inscribed triangle is a

diameter of the circle, then the triangle is a right triangle and the angle

opposite the diameter is a right angle)

- Theorem 10.11 (If a tangent and a chord intersect at a point on a circle, then the

measure of each angle formed is one half the measure of its intercepted

arc)

- Theorem 10.12: Angles Inside the Circle Theorem

- Theorem 10.13: Angles Outside the Circle Theorem

- Theorem 10.14: Segments of Chords Theorem

- Theorem 10.15: Segments of Secants Theorem

- Theorem 10.16: Segments of Secants and Tangents Theorem

- what tangents to a circle look like and how they exist to a group of circles

- the relationship between the measure of the central angle to the measure of the

major/minor arc

- the relationship between an inscribed angle and the measure of its intercepted arc

- standard equation of a circle and how it relates to the Pythagorean Theorem

Skills

Students will be able to...

- identify parts of circles (diameter, radius, center, tangent, secant, chord, points of

intersection, point of tangency, minor/major arcs, central angle, inscribed angle)

- draw and verify tangent lines to circles at a point of tangency

- construct common tangents to a group of circles

- find the measure of major/minor arcs based off of central angles and vice versa

- find the measure of inscribed angles based off the intercepted arc and vice versa

- describe how lines intersect a circle and give the number of intersections points in each

respective case

- find the length of segments of intersecting chords using the Segments of Chords

Theorem

- apply the Segments of Secants Theorem

- apply the Segments of Secants and Tangents Theorem

- write and graph equations of circles

- draw perpendicular bisectors and diameters using Theorem 10.4 and Theorem 10.5

- apply the Angles Inside/Outside the Circle Theorem

Stage 2: Assessment Evidence

Performance Tasks

- Description:

-Goal: The goal is for the students to understand how common tangent lines to circles

exist in the cases of 2 circles intersecting in 2,1, and 0 points.

- Role: The role of each student is to in groups find the tangent lines and form conjectures

on whether these findings would apply to all cases of interacting circles

- Audience: The audience is the teacher and other classmates

- Situation: The challenge is to demonstrate the construction of tangent lines to certain

groups of circles and form conjectures on how they would apply in all cases

- Purpose: Discovery in how tangent lines apply to any arbitrary circle or group of

arbitrary circles

- Standards: The performance by each group needs to involve correct construction of

tangent lines. Moreover, each group will have to give reasoning to their

conjectures and explain how they feel the conjectures would hold up in an

arbitrary case. The work will be graded by the attached rubric.

Participation will go into consideration of grade.

- Description:

- Goal: The goal is to landscape a yard with 3 bushes and find a point for a sprinkler that

is equidistant from each bush. This will be done in groups.

- Role: The students are landscapers and must plant the bushes and install a sprinkler in

the right position to equally water each plant.

- Audience: The audience will be hypothetical customers seeking results of their inquiry

- Situation: The challenge is to properly apply Theorems 10.4 and Theorem 10.5 to find

the equidistant point of the 3 bushes.

- Product: A sketch yielding the results of where the bushes and sprinkler should go

- Standards: The product must follow Theorem 10.4 and Theorem 10.5. The final sketch

will be judged by the attached rubric. Participation will be taken into account.

Other Assessments

- Exit Slips on specific days (see daily plans) to see how students perceived

daily lessons and new material.

- homework (see daily plans)

- worksheets and daily notes sheets (attached)

- 3 quizzes

- Unit test

Stage 3: Learning Plan Where are your students headed? Where have they been? How will you make sure the students

know where they are going?

This unit follows a unit on triangles. Students may know some properties about circles,

but this unit will take any knowledge and expand upon it in many ways. Students will

learn about tangents, secants, and most importantly, the equation of a circle and how to

graph them. This information is vital for Algebra 2, trigonometry, and for state testing.

How will you hook students at the beginning of the unit?

The students will be hooked in a few different ways. First the presentation of the

essential questions is the first part of the unit. They are not normal essential questions;

they involve real life applications and extrapolations of most of the knowledge they have

so they will highly interest the students. Secondly, the unit will begin with a real life

problem where the students will have to group up and brainstorm to think of a way to

solve the problem. Together these two tactics will hook the students.

What events will help students experience and explore the big ideas and questions in the unit?

How will you equip them with needed skills and knowledge?

With each new lesson, students will receive a notes sheet where they will fill in blanks of

important vocabulary and theorems. Also, students will do examples that illustrate the

main points of the lessons. Students will have homework throughout the unit and will

also have two performance tasks to complete. These performance tasks are designed to

make students think outside of the box and apply new knowledge to a real life problem.

Each section is full of theorems and techniques that will equip students in finding and

solving for the many applications of circles.

How will you cause students to reflect and rethink? How will you guide them in rehearsing,

revising, and rethinking their work?

While going over homework, students will be given the opportunity to ask questions

about challenging problems. In trying to get students to rehearse their work, students will

be given the opportunity to put problems on the board where they will have to explain

their thought processes and work to the other students.

How will you help students to exhibit and self-evaluate their growing skills, knowledge, and

understanding throughout the unit?

After many of the lessons, students will be given exit slips to complete. These exit slips

are designed so students can write down any questions/concerns that arose during the

introduction of the new material. They will cause students to evaluate themselves on

how well they understood that day's material. Homework will also serve as a self-

evaluation for the students. As the homework is checked, students will be able to see

how well they did. If they had trouble, they will be able to see it done either by the

teacher or other students where deep explanations will be given.

How will you tailor and otherwise personalize the learning plans to optimize the engagement

and effectiveness of ALL students, without compromising the goals of the unit?

(Accommodations and Modifications)

With each new lesson, students will be given the notes sheets where they have to fill in

blanks and do examples of the new material. This is done so students won't get behind at

all if they were taking notes. Also, students will be able to follow along and read while

the teacher reads over the notes so auditory and visual needs are met at the same time.

All of the examples will be done on a Promethean smart board or white board where

different colored markers can be utilized to differentiate between different points and

pieces that are being illustrated on the board. All assignments (homework, quizzes, test)

will be posted electronically so students can get copies in the event of an absence. For

students with IEPs, all necessary modifications are worked in to the unit plan so all

required modifications are covered. In engaging the students, there will be focus

questions throughout the unit with the addition of many real life problems where students

will be able to see some relevance into what they are learning.

How will you organize and sequence the learning activities to optimize the engagement and

achievement of ALL students?

See Learning Activities below.

*Notes:

1.) Graphics on notes sheets 10.1, 10.2, 10.3, 10.4, 10.5, 10.6, Focus Question-Quiz

10.1-10.2, Airplane Focus Question, and Focus Question #6 are taken from

McDougal Littell.

Learning Activities (by day)

Day 1:

Topic: Use properties of Tangents

Time: 42 minutes Objective: The objective of class is to introduce students to a new unit on circles, their

properties, and applications. The essential questions of the unit will be highlighted (at the

top of the unit plan) and will be posted so they are visible for the entirety of the unit. This

specific day will specifically introduce tangent lines and two theorems that arise from tangent

lines existing.

Understanding: Students will understand what a circle, tangent, secant, chord, diameter, and

radius are that they can identify them on a diagram. They will also begin to understand how

to work with the 2 theorems (10.1 & 10.2) they were introduced to.

Materials needed:

White board with markers

Promethean Smart Board

10.1 Properties of Tangents notes sheet (attached)

10.1 Exit Slip (attached)

Lesson 10.1, Practice A (attached)

Lesson Procedure:

1.) Class will begin with the teacher first announcing the beginning of a new unit on circles

and their properties. The essential questions will be read and posted on the wall. Any

questions regarding the essential questions will be answered at this time.

2.) Next, as a Focus Question, the teacher will pose a question to the class. Draw the earth

on the front board and locate two points- one on the equator and one at the north pole. The

question is, "If someone at the equator were trying to send a message to the north pole, what

path would the message follow?" Give the class a few minutes to think and collaborate.

After a few moments, bring the class back together. Ask for students to give some of their

thoughts; record some and ask why the students think the answers are right. If no one gets it,

show them how it would look (draw tangent lines from each pole and at their intersection put

a satellite). Explain that this is "line of sight" and these lines are actually tangent lines to the

earth.

3.) Hand out 10.1 notes. Go through worksheet filling in as the blanks come. The answers

can be found in the teacher's copy of the notes, which will be in a folder labeled Geometry.

4.) After finishing notes sheet, assign homework #1-17,24-33 (Lesson 10.1, Practice A).

5.) Finish class with the 10.1 exit slip.

Differentiation:

Students will receive handouts and will fill in blanks as the material is walked through. This

will meet auditory and visual needs. Students will also have opportunities to work in groups

as they answer the question from the beginning of the period.

Assessment:

The Exit Slip will give an early indication into how the material was delivered with any

questions directed back to the teacher. Homework will also serve as an assessment.

10.1 Properties of Tangents

Definitions

1.) A _________________ is the set of all points in a plane that are equidistant from a given

point called the _________________ of the circle.

2.) A segment whose endpoints are the center and any points on the circle is a ____________.

3.) A _________________ is a segment whose endpoints are on a circle.

4.) A _________________ is a chord that contains the center of the circle.

5.) A _________________ is a line that intersects a circle in two points.

6.) A _________________ is a line in the plane of a circle that intersects the circle in exactly one

point, the ______________________________.

Example 1:

Tell what the following are in relation to the picture.

a. segment AC:____________________

b. segment AB:____________________

c. line AE:________________

d. point C:________________

e. segment AF:_________________

Example 2:

Use the diagram to find the given lengths

a. radius of circle A=____

b. radius of circle B=____

c. diameter of circle A=_____

d. diameter of circle B=_____

Theorem

In a plane, a line is tangent to a circle if and only if the line is ____________________ to

a radius of the circle at its endpoint on the circle.

Example 3: Using the above Theorem, verify that line TS is a tangent to circle R.

Given: RS is a radius of circle R.

Theorem

Tangent segments from a common external point are ____________________.

Example 4: Using the above Theorem, solve for x.

x =_______

10.1 Exit Slip

1.) What were the main ideas of the discussion today?

______________________________________________________________________________

______________________________________________________________________________

______________________________________________________________________________

______________________________________________________________________________

2.) What questions/concerns do you have regarding the new material?

______________________________________________________________________________

______________________________________________________________________________

______________________________________________________________________________

______________________________________________________________________________

Day 2:

Topic: Use Properties of Tangents- Day 2

Time: 42 minutes

Objective: The objective of class is to first answer any questions/concerns that students posed

the day before on their exit slips. Next, the objective will be for students to form conjectures

on how tangent line to group of circles intersect (attached as Tangent Lines to Groups of

Circles). The last objective will be to cover homework and answer any developing questions.

Understanding: Students will understand how tangent lines are formed after going through the

performance task (see below). Also, they will understand the two theorems presented so far

(10.1 & 10.2).

Materials needed:


Promethean Smart Board

Tangent Lines to Groups of Circles sheet (attached)

Focus Question # 1 (attached)

Performance Task:

This task is titled Tangent Lines to Groups of Circles. To restate, students will be given a

few scenarios and directed to find tangent lines to groups of circles. In the examples, they

will be asked to explain how their findings would look in arbitrary cases. Finally, the second

essential question will be brushed upon.

Lesson Procedure:

1.) Begin class with Focus Question #1.

2.) Next go over any questions/concerns that the students showed on the exit slip from the

day before.

3.) Go over homework. Any problems that caused a lot of confusion put on board and

discuss as a class. For those problems that did not cause issue, have students put them on

board (should take about 20 minutes).

4.) Hand out the performance task and do as the sheet directs. Do this in groups of 4 or 5.

Hand out 1 rubric per group and make sure students understand what is expected.

Differentiation:

Students will be able to work in groups on the performance task. Also, any questions from

the exit slip from the day before will be answered to tailor to any specific needs. Finally, the

homework will be checked with any final questions being answered.

Assessment:

The performance task will be collected. As the students do this project, be watching for

cooperation by each group member; cooperation and participation will go into the grade.

Also, homework will serve as an assessment to see how well the students understood the

material.

Focus Question #1

1. Using the two equations below, solve for y.

5x-3y = 56

3x+y = 42

A. y = 3

B. y = 6

C. y = 2

D. y = 10

Tangent Lines to Groups of Circles

You and a team of mathematicians want to know how tangent lines exist to a group of two

circles. Look at the following examples and see if you and your team can formulate conjectures

on tangent lines to any group of two circles. You will present your findings as if you are

speaking at a math conference.

1.

2.

3.

These are just three example of an infinite number of ways in which two circles are in relation to

each other. With those around you, see if you can make a conjecture as to how many common

tangents would exist in the following cases.

Case 1: Two circles of any size not intersecting. How many common tangents would there be

and why?

______________________________________________________________________________

______________________________________________________________________________

______________________________________________________________________________

______________________________________________________________________________

______________________________________________________________________________

Case 2: Two circles of any size intersecting at one point (they are not overlapping). How many

common tangents would there be and why?

______________________________________________________________________________

______________________________________________________________________________

______________________________________________________________________________

______________________________________________________________________________

______________________________________________________________________________

Case 3: Two circles of any size overlapping but not intersecting. How many common tangents

would there be and why?

______________________________________________________________________________

______________________________________________________________________________

______________________________________________________________________________

______________________________________________________________________________

______________________________________________________________________________

This exploration was done in 2 dimensions. How would this same exploration look if the circles

were actually spheres and we were working in 3 dimensions? We will do this as a class.

______________________________________________________________________________

______________________________________________________________________________

______________________________________________________________________________

______________________________________________________________________________

______________________________________________________________________________

Tangent Lines to Groups of Circles Rubric

Names:______________________________________________________________________

1. Team cooperation and interaction

5 points

Students participated equally

in exploring and presenting

the data to the math

conference.

3 points

Some student involvement

was less than others. Only a

few members of the team

took all of the responsibility.

1 point

The majority of the group

was not involved. The work

that was done was only done

by one or two members.

2. Team presentation of findings.

5 points

The team presented their

findings in a professional,

clear, and succinct manner.

Their reasoning as to how

they got their findings was

clear and insightful.

3 points

The team presented their

findings in an average

manner. There were gaps

and some mistakes in the

reasoning.

1 point

The presentation gave no

explanation as to how the

students derived their data.

3. Team findings

5 points

The findings by the team

were correct in accuracy and

explanation.

3 points

The findings had a few

mistakes, but were correct

for the most part.

1 point

Team findings were not

correct and showed no signs

of insight.

Total:__________ _____/15

Day 3:

Topic: Find Arc Measures

Time: 42 minutes

Objective: The first objective is to go over the task that the students began the day before.

Students will present their findings to the class. The next objective is to introduce the new

lesson on finding arc measures to the students. Within the lesson, the students will be

learning about congruence of circles and arcs.

Understanding: Students will understand how arc measures are related to central angles. They

will also understand how to add measures of arcs and look at criteria to see if circles or arcs

are congruent.

Materials Needed:


Promethean smart board

10.2 Find Arc Measures notes sheet (attached)


Lesson Procedure:

1.) Begin class with students getting into their groups to finish the task from the day before.

Allow 10 minutes. Then have students present their findings to the class as if the class were

a conference listening to a team presenting new discoveries.

2.) Next, hand out 10.2 notes sheet. Go through and fill in as blanks come. The answers are

found in the Geometry folder.

3.) Assign homework #1-3, 4-20 odds, 22-24, 29-32, 33, 36(Lesson 10.2, Practice A).

Differentiation:

The students will be given notes and will fill in blanks as a class. This will meet visual and

auditory learning needs. They will also have the opportunity to pose questions to meet

specific needs.

Assessment:

Homework will serve as an assessment. Also, assessing student questions will give a look

into how well the students received the lesson.

Lesson 10.2 Find Arc Measures

Definitions

1.) A _____________________ of a circle is an angle whose vertex is the center of the circle.

2.) If the measure of an angle is less than 180o, then the points on the circle in the interior of the

angle form a ________________________.

3.) The points that do not lie on the minor arc form a _______________________.

4.) A ______________________ is an arc with endpoints that are the endpoints of a diameter.

Arc Addition Postulate

The measure of an arc formed by two adjacent arcs is the ___________ of the measure of

the two arcs.

Measuring Arcs:

The measure of a minor arc is the measure of its ___________________.

The measure of a major arc is the difference between 360o and the measure of the

related ______________________.

Example 1:

Find the measures of the following arcs:

a. Arc AB=_______

b. Arc ADB=_______

Example 2:

Use the Arc Addition Postulate to find the following arc measures:

a. Arc BD=_______

b. Arc BCD=________

c. Arc AD=_______

Congruent Circles and Arcs

Two circles are ____________________ if they have the same _____________. Two

arcs are ________________ if they have the same measure ad they are arcs of the

_____________ circle or of _________________ circles. If arcs or circles are congruent, you

write arc XY arc AB.

Example 2:

Tell whether the following arcs are congruent or not. Explain:

a. arc BC and arc DE

_____________________________

_____________________________

_____________________________

_____________________________

b. arc AB and arc CD

_____________________________

_____________________________

_____________________________

_____________________________

c. arc FG and arc HJ

____________________________

____________________________

____________________________

____________________________

Day 4:

Topic: Find Arc Measures- Day 2

Timing: 42 minutes

Objective: The objective of class will be to initially show how arc measure only relates to the

measure of the corresponding central angle. Aspects of a circle, such as circumference, area,

diameter, etc., do not affect the measure of an arc. The next objective will be to look over

homework and address any of the problems that caused any confusion.

Understanding: The students will understand that in finding arc measures, only the central

angles need to be considered. Next, the students will understand that one can find a measure

of an arc if a congruent arc has a given measure. Also, the students will understand that if an

arc is made of 2 smaller arcs, one can add arc measures to get a bigger arc measure.

Materials Needed:

http://www.mrperezonlinemathtutor.com/G/4_1_Central_Angles_and_Arcs.html

Promethean Smart Board & Computer


Focus Questions #2 (attached)

Lesson Procedure:

1.) Begin class with Focus Questions #2.

2.) Pull up

http://www.mrperezonlinemathtutor.com/G/4_1_Central_Angles_and_Arcs.html

Scroll down to the "Interactive Geometric Applets: Relevant Theorems." part and do the

interactive tutorial which illustrates the arc measure changing yet staying equal to the central

angle as it also changes. Discuss how these two parts are directly related.

3.) Draw a few circles on the board of different size. Assign arbitrary radii length to each.

Draw a 60 degree angle in each circle and show how the radius, diameter, circumference, etc.

do not affect the arc measure- only the central angle affects this.

4.) Go over homework. With those problems that caused confusion, have any students that

got it put it on the board and explain to the class how and why they did what they did to solve

the problem. If no one understood it, walk them through it.

5.) Announce there will be a short quiz the next day on the first two sections.

Differentiation:

Students will be able to see the tutorial which will allow them to see a circle's central angle

changing and arc measure changing together as the size is changed. All the work will be

done on the board with color-coding to meet specific visual needs. The homework will be

gone over in such a way that students can hear other ways of solving problems instead of

only hearing the teacher's approach.

Assessment:

Homework will serve as the main assessment.

Focus Question #2

2. Describe the solution to the system of equations below.

3x-y = 6

9x-3y = 18

A. The system has no solution.

B. The system has the unique solution (-6,-24).

C. The system has infinitely many solutions of the form y =3x-6

where x is any real number.

D. The system has the unique solution (2,0).

Day 5:

Topic: Quiz and Applying Properties of Chords

Timing: 42 minutes

Objective: The objective will be to assess the student's knowledge of the first two sections by

taking a short quiz (attached). The next objective will be to introduce the next topic. The

objective of the new topic is to learn about congruent chords and their arc measures,

perpendicular bisectors of chords, and chords equidistant from the center of a circle.

Understanding: Students will begin to understand the four theorems (10.3, 10.4, 10.5, 10.6).

Specifically, they will understand that in congruent circles or the same circle, two chords are

congruent if and only if they are equidistant from the center. Also they will understand that

in the same or congruent circles, two minor arcs are congruent if and only if their

corresponding chords are congruent. They will also understand that if a chord is a

perpendicular bisector of another chord, then the first chord is a diameter. Finally, they will

understand that if a diameter of a circle is perpendicular to a chord, then the diameter bisects

the chord and its arc.

Materials Needed:

Focus Question: Quiz 10.1-10.2 (attached)

Quiz 10.1-10.2 (attached)

10.3 Apply Properties of Circles notes sheet (attached)



Lesson Procedure:

1.) Class will begin with the Focus Question. It will be on the board when the students walk

in. It needs to be done quickly so the students can get to the quiz.

2.) Have students do quiz (should not take more than 15-20 minutes)

3.) Hand out 10.3 notes sheet and fill in blanks and examples as they come. Answers are in

the Geometry folder.

4.) Assign homework #1-25 odds (Lesson 10.3, Practice A).

Differentiation:

The focus question has two problems that are very similar to the problems on the quiz so the

work in solving the quiz problems should be fresh in the students' minds. The notes sheet

has blanks to fill in and will service auditory and visual learners as the material is introduced.

Assessment:

The quiz will serve as a summative assessment of the first two lessons.

Focus Question-Quiz 10.1-10.2

1. Use the diagram to find the measure of the radius of circle C and diameter of circle D.

Radius of circle C=_____________

Diameter of circle D=___________

2. Segment RS is tangent to circle C and segment RT is also. Find the value of x.

x=_______

Quiz: 10.1-10.2 Name:___________________

Use the following diagram to answer the following.

Given: Line GF and line GB are tangent to circle A

1.) Segment AF is a ______________.

2.) Line DC is a ______________.

3.) Point F is a point of ________________________.

4.) What is the measure of arc BH? _______________

5.) What is the measure of arc HF?________________

6.) What is the measure of arc FB?________________

7.) Segment EF is a ______________.

8.) How many units is radius AB?_____________

9.) On segment BG, what is x equal to? x=________

10.) What do you call a man who spent all summer at the beach?_______________

10.3 Apply Properties of Circles Recall

1.) A ____________________ is a segment with endpoints on the circle.

2.) A ____________________ is a chord that contains the center of a circle.

3.) A chord or diameter ________________ a circle into two ________________________.

Theorem

In the same circle, or in congruent circles, two minor arcs are congruent if and only if

their corresponding chords are congruent.

arc AB arc CD if and only ______ ______

Example 1:

Given: circle A circle D

segment BC segment EF

arc measure of arc EF=125o

Use the above Theorem to

find the arc measure of arc BC

____________

Theorem

If one chord is a perpendicular bisector of another chord, then the first chord is a

diameter.

In diagram, if segment QS is a perpendicular

bisector of segment TR, then segment ________

is a diameter of the circle.

Theorem

If a diameter of a circle is perpendicular to a chord, then the diameter bisects the chord

and its arc.

In diagram, if segment EG is a diameter and segment

EG is perpendicular to segment DF, then

segment HD segment HF and ________ _________.

Example 2:

Find the measure of segment BD. BD=________

Explain what theorem you used and why.

_________________________________

_________________________________

_________________________________

_________________________________

Theorem

In the same circle, or in congruent circles, two chords are congruent if and only if they

are equidistant from the center.

In diagram, segment AB segment CD

if and only if ________ ________.

Example 3:

In circle F, AB=CD=12. Find EF.

Day 6:

Topic: Apply Properties of Chords Day 2

Timing: 42 minutes

Objective: The objective will be to see how the presentation of the new section went the day

before. A very important part will be in proving the 4 theorems from the section. A final

objective will be to introduce the next performance task.

Understanding: The students will be able to involve themselves in proving the theorems with

the aid of the teacher. They will understand the pieces to the proofs because the steps will

pull from information they have learned prior to the lesson. They will also have a better

understanding of the information presented the day before.

Materials Needed:


Focus Question #3 (attached)

Designing Landscaping sheet (attached)


Performance Task:

This task is titled Designing Landscaping. Students will be given the scenario where they

have a landscaping business. They have to plant three bushes and install a sprinkler that

sprinkles the three bushes equally. To do this, they have to properly use theorems from the

present section. The rubric is attached.

Lesson Procedure:


2.) Class will begin by proving Theorems 10.3-10.6. It will be led by the teacher with as

much student interaction as possible. Have the students record the proofs on a piece of

paper.




4.) Hand out Designing Landscaping sheet. If time allows, let the students work on in

groups. If they do not finish, it is their homework.

Differentiation:

The homework will be gone over in a way where students can give their own ways of

thinking through problems which allows for different approaches to constructing knowledge.

Also, difficult problems will be done on the board so those with trouble can watch how they

are done.

Assessment:

Homework will serve as assessment and involvement in proving the theorems will act as a

personal assessment for the teacher to see how the well the students are remembering

important information.

Focus Question #3

3. Find the solution to the system of equations below.

4x-1y = 6

16x-4y = 18

A. x = -2, y = 1

B. x = 1, y = 1

C. x = -6, y = 2

D. x = 4, y = 2

Designing Landscaping

You have your own business in landscaping. A client has come to you asking you to

landscape a section of their front yard. They want three rhododendrons planted in front of their

house and want a sprinkler installed that will equally water the three plants. They know where

they want the bushes but don't know where to locate the sprinkler. Where would you install the

sprinkler? Why would you install it in the place that you did?

______________________________________________________________________________

______________________________________________________________________________

______________________________________________________________________________

______________________________________________________________________________

______________________________________________________________________________

______________________________________________________________________________

House

Designing Landscaping Rubric

Name:_______________

1. Sprinkler location

5 points

Sprinkler location is where it

will equally water all of the

bushes.

3 points

Sprinkler location will water

all of them but not equally.

1 point

Sprinkler location will not

water all three at once.

2. Work in finding sprinkler location

5 points

The work was correct in

finding the position to install

the sprinkler.

3 points

The work contained a few

errors in finding the sprinkler

location.

1 point

The work was wrong and

yielded the wrong spot for

the sprinkler.

Total:__________ ______/10

Day 7:

Topic: Use Inscribed Angles and Polygons

Timing: 42 minutes

Objective: One objective is to go over the Designing Landscapes sheet from the day before.

Another objective will be to introduce a new section to the students. The section focuses on

measuring inscribed angles, inscribed polygons, and circumscribed circles.

Understanding: The students will understand that the measure of an inscribed angle is one half

the measures of its intercepted arc. Also, they will understand that two inscribed angles

intercepting the same arc are congruent. Finally, they will understand that if a right triangle

is inscribed in a circle, then the hypotenuse is the diameter of the circle.

Materials Needed:

10.4 Use Inscribed Angles and Polygons notes sheet (attached)





Lesson Procedure:

1.) Begin class by having students get their Designing Landscapes sheet out from the day

before. Have them get into groups and have them exchange ideas. After 5 or so minutes,

bring the class back together. Ask for some ideas on how they solved the problem. Have

students hand in when they are done.


the Geometry folder.

3.) Assign homework #1-31 odds, 32 (Lesson 10.4, Practice A).

4.) At end of class, have students do exit slip and hand in.

Differentiation:

The students will be allowed to work in groups on the Designing Landscapes sheet in an

attempt for groups to formulate better ideas and understanding of the subject. Also, the notes

will be taken in such a way that the students follow along while filling in blanks to address

any visual or auditory needs. The pass out of class will meet specific needs because students

will get to write down any points that confused them or made them question the material.

Assessment:

The Designing Landscapes sheet will act as an assessment. Not only will the work on the

sheet be graded, but participation in group and class discussion will be taken into account.

The exit slip will also serve as an informal assessment where the teacher can see what points

were confusing to the students during lecture.

10.4 Use Inscribed Angles and Polygons

Definitions

1.) An ______________________ is an angle whose vertex is on a circle and whose sides

contain chords of the circle.

2.) The arc that lies in the interior of an inscribed angle and has endpoints on the angle is called

the ________________________ of the angle.

3.) A polygon is an ________________________________ if all of its vertices lie on a circle.

4.) The circle that contains the vertices of an inscribed angle is a

__________________________.

The Measure of an Inscribed Angle Theorem

The measure of an inscribed angle is one half the measure of its intercepted arc.

m ADB= 1/2 m arc AB

Example 1:

Find the following measures using the diagram

a.) m RST=________

b.) m arc QS=________

Constructing own knowledge:

Find the measure of arc HJ and angle HGJ. What do you notice about HGJ and

HFJ?

arc HJ = ______

HGJ = ______

Theorem

If two inscribed angles of a circle intercept the same arc, then the angles are

____________.

Proof:

Given: Circle O with inscribed C and D both intercepting arc AB.

1.) By the theorem about inscribed angles and their

intercepted arc, the measure of ACB is

_____________.

2.) Also the measure of ADB is _____________.

3.) Thus by the _____________________________,

we have ____________________, which implies __________________.

Theorem

If a right triangles is inscribed in a circle, then the hypotenuse is a diameter of the circle.

Conversely, if one side of an inscribed triangle is a diameter of the circle, then the

triangle is a right triangle and the angle opposite the diameter is the right angle.

The measure of ABC=_______ if

and only if _________ is a diameter of the circle.

Theorem

A quadrilateral can be inscribed in a circle if and only if its opposite angles are

supplementary.

Points D, E, F, and G lie on circle C if and

only if ______+______=______+______=_______.

10.4 Exit Slip


______________________________________________________________________________

______________________________________________________________________________

______________________________________________________________________________

______________________________________________________________________________


______________________________________________________________________________

______________________________________________________________________________

______________________________________________________________________________

______________________________________________________________________________

Day 8:

Topic: Use Inscribed Angles and Polygons Day 2

Timing: 42 minutes

Objective: The first objective is to cover any of the questions posed the day before on the exit

slip. Next, the objective will be to go over the homework and answer any

questions/examples that were difficult for the students.

Understanding: The students will understand how to apply the Measure of an Inscribed Angle

Theorem. Also, they will know that if 2 inscribed angles of a circle intercept the same arc,

then the angles are congruent. In regard to Theorem 10.9 and Theorem 10.10, students will

understand properties of inscribed triangles and polygons in a circle.

Materials Needed:




Lesson Procedure:

1.) Begin class with focus question #4 and answer any questions that the students posed the

day before on their exit slip.




3.) Do proofs for Theorem 10.9 and Theorem 10.10 (Example 30: Page 678) with the

students. Get them involved as much as possible; teacher will have to lead a lot of the

discussion but student involvement is key.

Differentiation:

Answering questions from the previous day's exit slip will individually meet student

questions. The homework will be gone over in a way where students can give their own

ways of thinking through problems which allows for different approaches to constructing

knowledge. Also, difficult problems will be done on the board so those with trouble can

watch how they are done.

Assessment:

Homework will act as an assessment of how well the students understand the material. Also,

involvement in proving the theorems will give an indication of how well students remember

and understand important theorems and other material from earlier lessons.

Focus Question #4

Simplify: (2ab)4

A. 8ab4

B. 12a5b5

C. 16a4b4

D. 8a4b4

Day 9:

Topic: Apply Other Angle Relationships in Circles

Timing: 42 minutes

Objective: The objective is to introduce a new lesson on applying other angle relationships in

circles. These other angle relationships involve chords, tangents, and chords intersecting in

unique ways inside or outside circles. Also with the focus question, another objective will be

for students to see the significance of learning the new material.

Understanding: Students will introduced to and begin to understand the Angles Inside the

Circle Theorem and Angles Outside the Circle Theorem. They will be able to find the

measures of angels in cases of chords, tangents, and secants intersecting in unique points in

or outside circles.

Materials Needed:



Airplane Focus Question (attached)

10.5 Apply Other Angle Relationships in Circles notes sheet (attached)


Lesson Procedure:

1.) Class will begin with the airplane focus question. The students will be allowed to

converse with neighbors to see if they can come up with any ideas. They won't learn how to

do the problem until later in class. Let them work for 5 minutes or so, hear their ideas, and

tell them you will come back to it later in class.


the Geometry folder

3.) Bring out problem from beginning of class again. They should be able to figure it out

with some help because the main theorem they will need they were given on the notes sheet.

4.) Assign homework #1-21 odds, 22, 23(Lesson 10.5, Practice A).

Differentiation:

The notes will be taken in such a way that the students follow along while filling in blanks to

address any visual or auditory need. Also, allowing the students to work in groups on the

airplane focus question may help initiate new ideas or approaches that students can learn

from each other.

Assessment:

In returning to the airplane focus question at the end of class, the teacher will be able to

gauge how the new material was received by seeing if the students can apply what they just

learned to a problem that they were not able to solve at the beginning of class. This is more

of an assessment for the teacher.

Airplane Focus Question

You are flying in an airplane about 5 miles above the ground. What is the measure of arc

BD, the part of the Earth you can see? (Earth's radius is about 4,000 miles)

How do you think we would solve this?

Meet in a group and see if you can come up with any ideas.

Given: lines CB and CD are tangents. This shows that segment CB ___ segment AB and

segment CD____ segment AD. From an earlier theorem, we know that _____ ______. Then

ABC ADC (by the H-L Congruence Theorem) and BCA DCA.

So m BCD 2(87.1o) = _______.

Let m BD = xo

Then m BCD = 1/2 (_________-_________)

______ = 1/2 [(360o- x

o) - x

o)]

xo ≈ ________

Thus, from the airplane, you can see an arc of about ________.

10.5 Apply Other Angle Relationships in Circles

Theorem:

If a tangent and a chord intersect at a point on a circle, then the measure of each angle

formed is one half the measure of its intercepted arc.

m 1=1/2 m __________

m 2=1/2 m __________

Example 1: Line m is tangent to the circle. Find the indicated measure

a. m 1= ________

Theorem: Angles Inside the Circle Theorem

If two chords intersect ___________ the circle, then the measure of each angle is one half

the ___________ of the measures of the arcs intercepted by the angle and its vertical

angle.

m 1 = 1/2 (m_______+ m________)

m 2 = 1/2 (m_______+ m________)

Example 2:

Secant FH and secant GJ intersect inside the circle. Find x.

x=_______

Theorem: Angles Outside the Circle Theorem

If a tangent and a secant, two tangents, or two secants intersect ____________ a circle,

then the measure of the angle formed is one half the _____________ of the measure of

the intercepted arcs.

1. m 1 = 1/2(________-________)

_____________ & ____________ intersect outside circle

2. m 2 = 1/2(________-________)

____________ & ____________ intersect outside circle

3. m 3 = 1/2(________-________)

____________ & ____________ intersect outside circle

Example 3:

The tangent GF and the secant GJ intersect outside the circle. Find x.

x =________

Day 10:

Topic: Apply Other Angle Relationships in Circles Day 2

Timing: 42 minutes

Objective: The main objective of class is to go over homework.

Understanding: Students will understand how to find angle measures in circles that have

intersecting chords, tangents, and secants.

Materials Needed:




Lesson Procedure:

1.) Begin class with Focus Question #5. It should only take about 5 minutes.




3.) Announce that there will be a quiz the next day on sections 10.3-10.5.

Differentiation:

The problems on the focus question are two similar problems to those that will be on the quiz

the next day. Also, the homework will be gone over in a way where students can give their

own ways of thinking through problems which allows for different approaches to

constructing knowledge. Also, difficult problems will be done on the board so those with

trouble can watch how they are done.

Assessment:

The focus question will give a good indication as to how the students are remembering the

previous 3 lessons. Also, homework will act as an assessment of how well the students

understand the material.

Focus Question #5

1.) Use the diagram to find x

2.)

a.) Find m DCE

b.) Find m arc DE

Day 11:

Topic: 10.3-10.5 Quiz and Find Segment Lengths in Circles

Timing: 42 minutes

Objective: The first objective is to get introduced to a new lesson. The lesson focus is dealing

with segment lengths in circles. Three important theorems are highlighted: the Segments of

Chords Theorem, the Segments of Secants Theorem, and the Segments of Secants and

Tangents Theorem. The final objective is to take the quiz on sections 10.3-10.5.

Understanding: Students will understand that if 2 chords intersect in a circle, then the product

of segment lengths of one chord is equal to the product of segment lengths of the other chord.

Also, they will understand that if two secants share the same endpoint outside of a circle,

then the product of the lengths of one secant segment and its external segment equals the

product of the lengths of the other secant segment and its external segment. They will also

understand that if a secant segment and a tangent segment share an endpoint outside a circle,

then the product of lengths of the secant segment and its external segment equals the square

of the length of the tangent segment. In understanding these three main points, they will also

understand how to apply these to different problems depending on the situation each problem

poses.

Materials Needed:

10.3-10.5 Quiz (attached)


10.6 Find Segment Lengths in Circles notes sheet (attached)

Lesson 10.6, Practice B (attached)

Lesson Procedure:


Geometry folder. This needs to take about 20 minutes because of a quiz.

2.) Have students take the quiz. They should be able to finish in the remaining 20 minutes.

3.) Assign homework #1-27 odds, 28-30 (Lesson 10.6, Practice B).

Differentiation:

The notes will be taken in such a way that the students follow along while filling in blanks to

address any visual or auditory needs.

Assessment:

The 10.3-10.5 quiz will act as a summative assessment.

10.6 Find Segment Lengths in Circles

Definitions:

1.) When two chords intersect in the interior of a circle, each chord is divided into two segments

that are called __________________________.

2.) A _____________________ is a segment that contains a chord of a circle and has exactly

one endpoint outside the circle.

3.) The part of a secant segment that is outside the circle is called an

_______________________.

Theorem: Segments of Chords Theorem

If two chords intersect in the interior of a circle, then the product of the lengths of the

segments of one chord is equal to the product of the lengths of the segments of the other

chord.

EA* ______= EC*______

Example 1:

Find ML and JK.

NK* NJ=______*______

x * (x + 5) = (_________) * (_________)

x2 + 5x = _________________

x = ________

Then ML = ________

And JK = ________

Theorem: Segments of Secants Theorem

If two secant segments share the same endpoint outside a circle, then the product of the

lengths of one secant segment and its external segment equals the product of the lengths

of the other secant segment and its external segment.

EA * ______ = EC * ______

Example 2:

Find x.

x =_______

Theorem: Segments of Secants and Tangents Theorem

If a secant segment and a tangent segment share an endpoint outside a circle, then the

product of the lengths of the secant segment and its external segment equals the square of

the length of the tangent segment.

EA2 = _______ * _______

Example 3:

Find RS.

RS = _______

Day 12:

Topic: Find Segment Lengths in Circles Day 2

Timing: 42 minutes

Objective: The objective of class is to go over the homework assigned the day before and

clarify any questions that the students pose.

Understanding: Students will be able to apply the three theorems from the section. They will

also know how to prove the Segments of Chords Theorem.

Materials Needed:




Lesson Procedure:

1.) Begin class with Focus Question #6. Give students about 5 minutes to finish.

2.) Go over proof of Segments of Chords Theorem.




Differentiation:

The focus question involves a real-life problem so the students will be able to see some

relevance in what they are learning. Also, the homework will be gone over in a way where

students can give their own ways of thinking through problems which allows for different

approaches to constructing knowledge. Also, difficult problems will be done on the board so

those with trouble can watch how they are done.

Assessment:

The focus question and homework will be good indicators of how well the new material was

received by the students. Also, as the Segments of Chords Theorem is gone through, the

teacher will get a good indication of how well the students are remembering important facts

from previous lessons.

Focus Question #6

You are standing at point C, 45 feet from the Point State Park fountain in Pittsburgh, PA. The

distance from you to a point of tangency on the fountain is 105 feet. Find the distance CA

between you and your friend at point A (Hint: Use Segments of Secants and Tangents Theorem).

Solution:

Day 13:

Topic: Write and Graph Equations of Circles

Timing: 42 minutes

Objective: The objective is to introduce the final lesson of the unit. There are no theorems to

learn or remember in this lesson, but learning the equation of a circle is extremely important.

Students will use the equation of a circle many times, and it will be crucial to know for their

many standardized tests.

Understanding: Students will understand how the equation of a circle is directly related to the

distance formula. They will also understand how to write the equation of a circle with any

origin and any radius. Not only will they know how to write they equation, but they will

know how to graph them as well.

Materials Needed:




10.7 Write and Graph Equations of Circles notes sheet (attached)



Lesson Procedure:


2.) Next, hand out 10.7 notes sheet and fill in blanks and examples as they come. Because

this is such an important section in mathematics, make sure students understand each step

and example clearly illustrated on the notes sheet.

2.) Assign homework #1-32 odds, 33-36 (Lesson 10.7, Practice A).

3.) Hand out exit slip and have students hand in when they leave.

Differentiation:

Also, the notes will be taken in such a way that the students follow along while filling in

blanks to address any visual or auditory needs. The exit slip will meet specific needs because

students will get to write down any points that confused them or made them question the

material.

Assessment:

The exit slip will give an indication as to how the students received the new information.

Based off the student responses, teaching can be changed the next day to answer or clarify on

any questions that the students posed.

Focus Question #7

Simplify: (22a3b)2· (23ab2)2

10.7 Write and Graph Equations of Circles

Circles Centered at the Origin

Let (x, y) represent any point on a circle with

center at the origin and radius r. By the

Pythagorean Theorem,

x 2 + y

2= r

2.

This is the equation of a circle with radius

r and center at the origin.

Example:

Write the equation of the circle.

Solution:

The radius is 3 and the center is the origin.

x 2 + y

2= r

2

x 2 + y

2= _____

So the equation is _________________.

Circles not Centered at the Origin

You can write the equation of any circle if you know its radius and the coordinates of its center.

Suppose a circle has radius r and center (h, k). Let (x, y) be a point on the circle. The distance

between (x, y) and (h, k) is r, so by the Distance Formula

Square both sides than to get the _______________________.

Standard Equation of a Circle

For center (h, k) and radius r, the standard equation of a circle is

(x - h) 2

+ (y - k) 2 = r

2

Example 2:

Write the standard equation of a circle with center (2, 7) and radius 4.

Example 3:

Graph the equation (x - 4) 2 + (y + 1)

2= 16.

10.7 Exit Slip


______________________________________________________________________________

______________________________________________________________________________

______________________________________________________________________________

______________________________________________________________________________


______________________________________________________________________________

______________________________________________________________________________

______________________________________________________________________________

______________________________________________________________________________

Day 14:

Topic: Write and Graph Equations of Circles Day 2

Timing: 42 minutes

Objective: The objective will be to solidify how to write equations of circles and draw them

given either the radius and center, or just the equation.

Understanding: The students will understand how to write the equation of a circle and graph

them.

Materials Needed:




Chapter 10 Review sheet (attached)

Lesson Procedure:


2.) Next, students will go over their homework. With those problems that caused confusion,

have any students that got it put it on the board and explain to the class how and why they did

what they did to solve the problem. If no one understood it, walk them through it.

3.) Announce there will be a quiz the next day on 10.6-10.7.

4.) Hand out the Chapter 10 review sheet and let students work on it until the class is over.

Differentiation:

The homework will be gone over in a way where students can give their own ways of

thinking through problems which allows for different approaches to constructing knowledge.

Also, difficult problems will be done on the board so those with trouble can watch how they

are done.

Assessment:

Homework will act as an assessment to see how the students are doing with the new

information. Also in reviewing for the quiz, the teacher will get a feel for how well the

students are learning and understanding the material.

Focus Question #8

Simplify: (3a2b3)·(22ab)2

Day 15:

Topic: 10.6-10.7 Quiz and Chapter Test Review

Timing: 42 minutes

Objective: The objectives will be to take the quiz and review for the chapter test the next day.

Materials Needed:

10.6-10.7 Quiz (attached)

Lesson Procedure:

1.) Students will take 10.6-10.7 Quiz. This should only take 20 minutes.

2.) Students will go over the Chapter 10 Review sheet from the day before.

Differentiation:

Giving the students the answer key to the review sheet will allow them to see how they are

doing.

Assessment:

Quiz 10.6-10.7 will be a summative assessment with the chapter test the next day also acting

as the same.

Day 16:

Topic: Chapter 10 test

Timing: 42 minutes

Materials Needed:

Chapter 10 test (attached)

Lesson Procedure:

1.) Students will take the Chapter 10 Test. It will take the entire period.

Differentiation: After the test is graded, students will have the opportunity to revise the

problems they got incorrect. For each problem they missed, they can receive half credit by

explaining how they got their new answer in a very explicit and descriptive way. Through

this explanation, the students will gain insight to why they answered the question incorrectly.

Assessment:

The test will act as the final assessment of the chapter.

Chapter 10 Test

Name:_____________

Period:______

Fill in the appropriate blanks.

1. In a plane, a line is ______________ to a circle if and only if the line is perpendicular to a _______________ of the circle at its endpoint on the circle.

2. If two ______________ angles of a circle intercept the same arc, then the angles are

congruent. 3. A _______________ of a circle is an angle whose vertex is the center of the circle. 4. If a tangent and a chord intersect at a point on a circle, then the measure of each angle

formed is _________ the measure of its _______________ arc. 5. Tangent segments from a common external point are _______________. 6. If a secant segment and ______________ share an endpoint outside a circle, then the

product of the lengths of the secant segment and its external segment equals the square of

the length of the __________________. 7. A quadrilateral can be inscribed in a circle if and only if its opposite angles are

__________________. 8. The measure of a minor arc is the measure of its _______________________. 9. If two chords intersect _______________ the circle, then the measure of each angle is ½

the sum of the measures of the arc intercepted by the angle and its ______________

angle's intercepted arc. 10. If two secants share the same endpoint ___________________ of a circle, then the

product of the lengths of one __________________ and its external segment equals the

product of the lengths of the other secant segment and its ____________________. 11. If one chord is a perpendicular bisector of another chord , then the first is a

_________________. 12. The measure of an ________________ angle is ½ the measure of its intercepted arc. 13. In the same circle, or congruent circles, the minor arcs are congruent if and only if their

_____________________ chords are congruent. 14. If a tangent and a secant, two tangents, or two secants intersect _________________ a

circle, then the measure of the angle formed is ½ the _________________ of the measure

of the intercepted arcs. 15. In the same circle, or congruent circles, two chords are congruent if and only if they are

___________________ from the center. 16. If a right triangle is inscribed in a circle, then the hypotenuse is a _______________ of

the circle. Conversely, if one side of an inscribed triangle is a ____________, then the

angle opposite the diameter is the ___________ angle.

17. If two chords _______________ in the interior of a circle, then the product of the lengths

of the segments of one chord is equal to the product of the lengths of the

______________ of the other _____________. 18. If a diameter of a circle is ____________ to a chord, then the diameter _____________

the chord and its arc.

19. True or False: A circle has 2 or more tangent lines at a unique point on a circle.

20. True or False: Two overlapping, but not intersecting circles, have 1 common tangent

line.

21. True or False: If two circles have one common tangent, then they must be overlapping

and intersecting at one point.

22. Write the standard equation of a circle with center (0,0) and a radius of 2.

23. Write the standard equation of a circle with center (-2,2) and a radius of

24. The three dots below are three different cities. A radio company wants to build an office

where they can equally transmit to each of the cities. Where should the company build

their office and why? Draw the location and explain how you got your answer.

Section 110.1Section 210.2Section 310.3Section 410.4Section 510.5Section 610.3-10.5 QuizSection 710.6Section 810.7Section 9Chapter 10 Test ReviewSection 1010.6-10.7 QuizSection 11Chapter 10 Testimg-4161117-0001.pdf

Documents

Geometry 16 Day Unit 42 Minute Classes Mr. Nathaniel Frye Beginning Date: March … · 2018. 10. 4. · -Theorem 10.1 (Line is tangent to a circle if and only if the line is perpendicular