unit 12: circle equations · unit 12: circle equations in this unit you must bring the following...

Name: ___________________________

Geometry Period _______

Unit 12: Circle Equations

In this unit you must bring the following materials with you to class every day:

Calculator Pencil

This Booklet

A device

Headphones!

Please note:

You may have random material checks in class

Some days you will have additional handouts to support your understanding of

the learning goals in that lesson. Keep these in a folder and bring to class every

day.

All homework for part one of this unit is in this booklet.

Answer keys will be posted as usual for each daily lesson on our website

Today's Goal: What is the equation of a circle? How do we graph/write equations for circles?

I. Launch. Imagine it: A. As a gift to the school, your class purchased a fountain for the front of the school. You also purchased 8 flower plants to

decorate the area near the fountain. What shape would be formed if you planted your flowers such that they were all

equidistant to the fountain?

II. Refine your Thinking! A. What is the name of the circle?

B. What do we call a segment such as the one from a to P? b to P?

C. How do they relate? Why/ how do we know this?

III. Transfer your thinking! Since all points are equidistant, let's find the distance they all are from the center point!

Example 1

Example 2

12-1 Notes

Example 1

Example 2

Big Take Away! Look at the two problems above...

1) What do you notice about the center point and how it's represented in the equation of a circle?

2) What do you notice about the radius and how it’s represented in the equation of a circle?

Example 3: Write the equation of a circle with center at (-3, 5) and a radius of 3.

Equation of a Circle with center (h,k) and radius r

a) At what point is the circle centered?

b) What is the radius?

a) At what point is the circle centered?

b) What is the radius?

Example 4 What is the center and radius of the following? 4x2+4y2=36

Note:

Example 5 - Graphing Circles

Graph the circle whose equation is: x2 + (y - 2)2 = 25

Use a compass for the

perfect circle!

12-1 Homework

Directions: Complete #’s 1-6 on one side. Each column helps you practice the same content, but the right column

allows you to challenge yourself! You can go back and forth if you want!

1) Graph and write the equation of a circle with center at the origin and a radius of 6.

2) 3) The student radio station is located at point (-1,-4) on a coordinate plane and has a broadcasting range of 20 miles. Write the equation that will represent the outer edge of the broadcasting range.

1) Graph and write the equation of a circle with center at

(-6,7) and a radius of √ . Hint: You’ll have to estimate the length of the radius.

2) A circle is represented by the equation x2+(y-a)2=b. What are the coordinates of the center of the circle and the length of the radius in terms of a and b.

3) The student radio station is located at point

(-10,4) on a coordinate plane and has a broadcasting

range of 2√ miles. Write the equation that will represent the outer edge of the broadcasting range.

4) In the accompanying diagram, the center of circle O is

, and the coordinates of point P are . If is a

radius, what is the equation of the circle?

5) Given the equation x2 + (y – 2)2 = 25 Determine if the point: (-1,7) is on the circle. Explain your answer.

6) On the coordinate plane sketch all points that are 3 units from the point (0,-1). Write the equation that satisfies the given condition.

1)

2)

3)

4)

4) The endpoints of a diameter of a circle are (4,-2) and (-2, -2). Which is the equation of this circle? 1)

2)

3)

4)

5) Given the equation x2 + (y – 2)2 = 25 Determine if the point: (-1,7) is on the circle. Explain your answer. 6) What is the equation of the locus of points equidistant from the point (-4,2), by a distance of 6? Sketch it

Today's Goal: How do we complete the square to re-write equations?

Group Flashback/discovery! In your groups, read and answer the questions that follow. Make sure you work together and that everyone agrees on

each answer before moving on!

1. Factor the following quadratics using the sum and product method:

a.

b.

c.

2. Examine the factors of each trinomial factored above. What do you notice? Is this true for

EVERY quadratic?

3. Re-write the factors in squared form .

a. b. c.

Perfect square trinomials

The types of quadratics you worked on in your groups are called

The factors of perfect square trinomials can always be we written in the form .

Why are these types of trinomials necessary?

12-2 Notes

Creating perfect square trinomials within an equation:

Example 1: Complete the square to create a perfect square trinomial.

Then write in factored form.

You try one!

Class discussion

What is different here in this equation?

Same procedure from before, but now we are creating __________ perfect square trinomials.

Let’s do it!

What does this form remind you of?.....

Procedure:

1. The coefficient of the squared

variable must be 1-if not divide by

the coefficient.

2. Move any numerical constants (plain

numbers) to the other side of equal

sign.

3. Get ready to insert the needed value

for creating the perfect square

trinomial. Remember to balance

both sides of the equation.

4. Find the missing value by taking

half of the "middle term" and

squaring. This value will always be

positive as a result of the squaring

process.

• Rewrite in factored form.

Procedure:

1. Same as steps 1-2 from

above. 2. Group common variables

together. 3. Find each missing value by

taking half of the "middle

term" and squaring. This

value will always be positive

as a result of the squaring

process. 4. Rewrite in factored form.

You try!

x2 + y2 – 6x + 4y – 3 = 0

Practice Directions: Complete all problems. Make sure you read each problem carefully before answering.

1. What values would be placed in the boxes to create perfect square trinomials?

a.

b.

c.

Complete the square to create perfect square trinomial/s and re-write them in factored form.

2. Write the center-radius equation of a circle with a center at (-3, -6) and passes through the point (-4, 8). Hint: How

might you be able to find the DISTANCE/length of the radius?

12-2 Homework

1. Match!

Match each graph with its equation:

2. Complete the square to create perfect square trinomial/s and re-write them in factored form.

a. x2 – 6x + y2 + 4y = 12

b.

Equation 1:

Matches Graph ________

Equation 2:


Equation 3:


Equation 4:


c. x 2 + y 2 − 2 x + 4 y + 1 = 0.

3. State the equation of a circle which has a center of (5,-3) and a radius of 9.

4. Find an equation of the circle whose center is at the point (-4 , 6) and passes through the point

(1 , 2). Hint: We need to find the DISTANCE of the radius.

Today’s Goal: How do you convert a circle equation to center-radius form?

With your teammates:

Use the equation shown right to answer #1-3:

1) Why does the following equation represent a circle?

2) Looking at the following equation, can you identify the center and/or radius of the circle?

3) What obstacles are getting in the way?

Read together, highlight as you read: The equation shown above IS a circle! We call this form the: STANDARD/GENERAL form of a circle.

When the equation of a circle appears in "general form", it is often beneficial to convert the equation to "center-radius"

form to easily read the center coordinates and the radius for graphing.

In order to convert the general form of a circle to the center-radius form, we use a technique called completing the

square. (LAST CLASS)!

Work together as a team and answer all parts:

a) Determine and state the center and radius of the following circle: Think back to yesterday!

0

b) Graph the circle from part a.

CHECK- IN WITH YOUR TEACHER PRIOR TO MOVING ON!

12-3 Notes

Continue in your team!

Example #1) Determine the center and radius of the following equation of a circle.

Example 2) Determine the center and radius ( in simplest radical form) of the following circle by completing the square:

x2 + 2x + y2 - 4y = 3

Example 3) The Equations of a circle is . What are the coordinates of the center and the length of

the radius of the circle?

What method did you use?

What’s different here?

Do I need to complete the

square twice?

Remember! Circles of the

form 𝑥 𝑦 𝑟 have a

center at the origin!

Example 4) Use the following general form for a-c:

a) Write the equation of the circle in center-radius form

b) Identify the center and radius of the circle

c) Graph the circle

Example 5) The point (3,4) is on a circle whose center is (1,4). Write the center-radius form of the circle. Use of the grid

is optional, but may be preferred…

Example 6) Show that is the equation of a circle. What is the center of this

circle? What is the radius of this circle?

Example 7) State the equation of a circle in center radius form which has a center at (5, -3) and a radius of 9.

Example 8) Write the general form equation for the circle whose graph is shown at the right.

Careful! spicy

12-3 HW

1. Are the following points on the circle graphed correctly? (Yes or no?)

*on doesn’t mean in -Careful!

a) (-3,2) ______ b) (1,-2) ______ c) (1,2) ______ d) (1,4) ______

2. Convert the equation to center-radius form. State the coordinates of the center of the circle and its radius.

x2 – 6x + y2 + 4y = 12

3. State the equation of a circle which has a center of (5,-3) and a radius of 9.

4. What is the center and radius of the following: 2x2 + 2y2 = 12

5. Write the center-radius equation of a circle with a center at (-3,-6) and passes through the point (-4,8). Use of the grid

is optional.

6. Show that the following equation is the equation of a circle. What is the center and radius of that circle? ( Hint:

Double-Distribute – Algebra Review)!

7.

Show ALL WORK

HERE!

Today's Goal: How do we solve Systems with lines and Circles?

Class Brainstorm! 1. What do we know about lines and linear equations?

2. What do we know about circles and circle equations?

3. What do you predict a line and circle system might be?

Reactivate our Knowledge: What is a System of equations? Solution to a System of Equations:

Let’s try it!

Find and state the solution to the following system of equations graphically.

Line Circle

How can we be sure of our solutions?

12-4 Notes

Practice time with a partner!

1. Find and state the solution for the following system of equations.

x – y = 3

(x – 2)2 + (y + 3)2 = 4

2. Solve the following system and check your answers. (HINT: USE A COMPASS FOR THIS ONE)!

x = 1

(x +3)2 + (y -1)2 = 25

Practice time on your own!

Directions: Solve the following systems of equations graphically:

3. y = –x – 3 x2 + y2 = 9

4. *Is (8,6) a solution to the system below? Explain why or why not using algebraic reasoning!

5. Solve the following system of equations.

6. Solve the following system of equations

x + y = -2

x2 + y2 -4x – 2y = 20

12-4 Homework

Directions: You should do all the questions on this assignment. Show your work.

Don’t’ forget to check your answers!!

1) Solve the following system of equations graphically and state the coordinates of all solutions.

Algebraically check your solutions here!

2)

3) A circle with the equation does not include points in Quadrant (sketch to see!)

1) I

2) II

3) III

4) IV

4) Is (-1,3) a solution to the following system? Explain why or why not using algebraic reasoning!

5) The equation of a circle is x2+ y2 -6y + 1 = 0. What are the coordinates of the center and the length of the radius of

this circle?

6) Graphically find the solution to the following system of equations:

3x2 + 3y2 = 48

4x2 + 4y2 = 144

Watch the assigned EDPUZZLE video of a previously learned concept. Answer the question that follows. Mastery of the content of this video is essential for our next lesson in class. Failure to watch the video will result in confusion and your inability to interact with your peers throughout the lesson. This page will be checked tomorrow in class and an entrance ticket into class will be assigned to prove your mastery of the concept.

a)

On your own…

b) Now graph line y = -1

c) State solutions

to this system of

equations.

NOTE: When picking your x-values for the

table, you don’t have to calculate the

vertex point; look for where your table of

values changes direction (do you see the

symmetry?)

Video: Graph the following parabola: y = x2 + 2x -1

Today’s Goal: What is the area of the sector of a circle?

Re-activate your knowledge with a shoulder buddy!

Define each term below and match the labeled part of the diagram to the term that it represents:

Radius:

Define: a segment from the _____ to

the edge of the cirlce.

Example:

Center:

Define: a point in the “middle” of a cirlce,

from which all distances to the edge of the circle

are the same.

Example:

NEW VOCAB ALERT!! Central Angle:

Define: an angle whose ____________ is the _________ of a circle,

and whose sides are the radii.

Example:

Arc:

Define: a ______________ or ____________ of a circumference of a circle.

Example:

Sector

Define: a part or piece of the inside of a circle, formed by the _______________ _________________.

Example:

Area of a Circle:

12-6 Notes

Diameter:

Define: a segment draw

across a circle passing

through the ________________

Example:

Are you familiar with

all of these words?

Make connections! What does it look

like?

Shoulder Buddy Exploration!

1: What is the area, in terms of , of a circle with radius of 6 inches?

2:The circle from question #1 was split up! What is the area of the shaded region in terms of ?

But first!!!!

How would you describe the shape of the shaded region? _________________________

How many degrees is the central angle? _____________________________

How many degrees is the full circle? ___________________________________

What fraction of the circle is the shaded region? This fraction is the same as

Now use the fraction from the previous question to solve for the area of the shaded region? Show all work!

3: The circle question #1, was split up again! What is the area of the shaded region in terms of ?

But first!!!!





Now use the fraction from the previous question to solve for the area of the shaded region? Show all work!

4: A little different! The circle from question #1, was split up again! What is the area of the shaded region in terms of ?





Think about questions 2 and 3! Now use the fraction from the previous question to solve for the area of the

shaded region? Show all work!

Make a prediction! What will be the general formula for area of a sector?

Back Together!

The area of a sector of a circle is:

Let’s try another!

Example 3: Determine the area of the acute sector formed by <UTV to the nearest tenth.

Time to Practice on your own!

4. Determine the radius, to the nearest inch, of a circle with an area of 380 square inches

5. Determine the area of circle V, given that the area of sector VTU is 35m2.

Error prevention!! An answer

could be rounded or left in

terms of 𝜋, what does that

mean?

6. Find the areas of both sectors (the small and big sector) formed by <DFE. Round your answers to the nearest

tenth.

7. The area of the shaded sector is shown. Find the area of circle M to the nearest whole number.

8. The diagram shows the area of a lawn covered by a water sprinkler. For parts a-c round to the nearest whole

number!

a. What is the area of the lawn that is covered by the sprinkler?

b. Uh-oh! The water pressure weakened! Now the radius is 12

feet! What is the area of the lawn that will be covered.

c. What is the area of your lawn that is no longer being watered?

12-6 Homework

Directions: Complete each of the following problems. Show all work to earn credit.

1. Solve for the area and circumference of a circle with a diameter of 6.

Leave in terms of .

2. The diagram shows a projected beam of light from a

lighthouse. For a and b round to the nearest tenth.

a. What is the area of water that can be covered by the light

from the lighthouse?

b. What is the area of land that can be covered by the light from the lighthouse?

3. The area of the shaded sector is shown. Find the radius of circle M to the nearest whole number.

water

land

4. Your friend claims that if the radius of a circle is doubled, then its area doubles too. Is your friend correct?

Show an example to prove your response.

5.

5. In the diagram below of circle O, diameter and radii and are drawn. The length of is 12 and the

measure of is 20 degrees. If the area of sectors AOC and BOD are equal, find the area of sector BOD in terms

of .

6. In the circle, O is the center. The radius of the circle is 7 meters. Find the area of the shaded region to the nearest 10th

.

Continue to next page!

12-6 Video Notes Watch the Assigned ED PUZZLE!

You will be quizzed on this tomorrow!

Learning Goals: What are key ideas for tomorrow’s lesson on arc length?

Briefly describe the following terms:

Circumference Sector Arc of a Circle

Quick Check!

Circumference: C =

Important Notation/Concepts:

“arc AB”- Determined by _____ ______ on the circle. Measured

in __________________

The measure of an arc is the SAME as the measure of the

_______ _________

MAJOR ARC -

Minor Arc -

Tomorrow’s lesson we focus on the ____________________ of an arc. That is the distance (in cm, ft,

inches, etc) of a part of the circumference.

Arc Length (Degrees)

Learning Goals: How do we find the measure of an arc of a circle? How do we calculate the length of an arc?

WHAT SHOULD THE AMUSEMENT PARK DO?

An amusement park has discovered that the brace that provides stability to the Ferris Wheel has been

damaged and needs work. The Ferris wheel has 12 seats supported by 12 beams (radii) that measure 14 feet.

The arc length of steel reinforcement that must be replaced is between the two seats shown below. What is

the length of steel that must be replaced to the nearest foot? Describe the steps you used to find your

answer.

Jot down any initial thoughts of what you think you might do to answer this question! What are we looking

for? What do we know? What might we need to calculate?

Initial Thoughts Solution ( WAIT! after class discussion)

12-7 Notes

LET’S COME TOGETHER!

Compare this to yesterday’s lesson!

What’s the similar? What’s different?

Shoulder Buddy Exploration!

Example 1)

a) What is the circumference, in terms of , of a circle with radius of 10 inches?

b) What is the circumference of the shaded region in terms of ? (All regions are cut evenly)

My thinking:

How would you describe the shape of the shaded region? ______________

How many degrees is the central arc? _____________________________

How many degrees is the full circle? _______________________________

What fraction of the circle is the arc of the shaded region? This fraction the same as

_____________

So, what is the arc length of shaded region in terms of π?

Make a prediction! What will be the general formula for length of an arc?

Circumference of a Circle:

Back Together!

Arc Length of a Circle:

Let’s try one!

The length of PA is 8 cm °. What is the length of the minor arc ̂ in terms of π?

How would your approach be different if we were asked for the length of the major arc ̂?

*Now, let’s go back to the Ferris Week question and help the amusement park figure out a solution to the problem!

Practice

1. The length of RT is 15.28 m and the measure of is 144˚. What is the length of major arc ̂ to the

nearest tenth of a meter?

2. The length of ̂ is 4.19 in and °. What is the circumference of circle Z? (Careful here!)

3. The dimensions of a car tire are shown. To the nearest foot, how far does the tire travel when it makes 15 revolutions?

5. Find the length of minor arc ̂ to the nearest tenth:

Determine the area of the sector to the nearest tenth. 6. Given circle A with equation x2 + y2 = 9.

a) Find the circumference of circle A. Leave your answer in terms of pi. b) What is the area of a sector of circle A with central angle measure of 35 degrees in terms of ?

12-7 Homework

Directions: Answer the following questions to the best of your ability. Show ALL of your work!!

1. What are the lengths of the major and minor arcs below in terms of π?

2. Express the circumference of circle R when the length of arc PQ is 3.82 m:

3. What is the circumference of the circle with equation (x+1)2 + (y-5)2 = 28? Round your answer to the nearest hundredth.

4. Match the term to the definition:

A) Sector The distance around a circle_____

B) Circumference A part of the circumference of the circle “crust” _____

C) Arc A part of the area of the circle “pizza slice” _____

5. In the diagram below of circle O, the area of the shaded sector LOM is .

If the length of is 6 cm, what is ?

1) 10º 2) 20º 3) 40º 4) 80º

Arc Length (Radians)

Learning Goals: (1) What is a radian? (2) How can I solve for arc length given an angle in radians?

Warm-Up: Answer (or attempt to answer) the following 2 questions. 1) 2)

What do you notice is the same about the two circles above?

What do you notice is different about the two circles above?

Based on what you see, what do you think radians are?

Solve: What is the length of the minor arc below to the

nearest whole number?

In a circle, a central angle of 3 radians intercepts an arc of

5 centimeters. Find the radius in centimeters to the

nearest 10th.

*What new vocab word do you see in one of the problems

that may be preventing you from solving it?

We can use a new way to solve ARC LENGTH questions involving RADIANS!

In general, if

____ is the measure in radians of a central angle

____ is the length of the intercepted arc, and

____ is the length of the radius, then:

Note: The angle MUST be in radians in order to use this formula!!

12-8 Notes

Watch out! “r” is

“radius”

NOT radians!

Let’s try some examples!

1. In a circle, the length of a radius is 4 centimeters. Find the length of an arc intercepted by a central angle whose measure is 1.5 radians.

Steps: Solution:

1) Formula first

2) Substitute the given values.

3) Solve for ________

Answer:

Let’s go back to the warm-up problem and solve it using our new method!

In a circle, a central angle of 3 radians intercepts an arc of 5 centimeters. Find the radius in centimeters to the nearest 10th Today’s practice: Remember to identify if you are working with radians or degrees before you solve each question! 1. If S = 12 and r = 4, find . ***INCLUDE YOUR UNIT IN YOUR FINAL ANSWER!

2. A wedge-shaped piece is cut from a circular pizza. The radius of the pizza is 6 inches. The rounded edge of the crust of the piece measures 4.2 inches. To the nearest tenth, the angle of the pointed end of the piece of pizza, in degrees, is…

3. A sprinkler system is set up to water the sector shown in the accompanying diagram, with angle ABC measuring 1 radian and radius AB = 20 feet.

a) What is the length of arc AC, in feet?

b) What is the length of arc AC with angle ABC measuring

radians and radius AB = 20 feet, to the nearest 10th.

4. If the central angle of a circle is 47 degrees and r = 3 units, find arc length to the nearest 100th.

5. If = 2

1 and r = 3mm, find S.

6. A dog has a 20-foot leash attached to the corner where a garage and a fence meet, as shown in the accompanying diagram. When the dog pulls the leash tight and walks from the fence to the garage, the arc the leash makes is 55.8 feet.

a) What is the measure of angle between the garage and the fence, in radians?

b) Express your answer from part (a) in degrees to the nearest degree.

12-8 Homework 1. What is the value of the arc length, s, in the circle below in terms of ? *Note this angle is in radians

2. Jack wants to plant a border of flowers in the shape of an arc along the edge of a circular walkway. If the circle has a

radius of 5 yards and the angle subtended by the arc measures radians, what is the length, in yards, of the

border? 1) 0.5

2) 2

3) 5

4) 7.5

3. A ball is rolling in a circular path that has a radius of 10 inches, as shown in the accompanying diagram. What distance has the ball rolled when the angle subtending the arc is 54°? Express your answer to the nearest hundredth of an inch.

4. Nick has to solve for the radius in the problem shown below. Identify and correct Nick’s two mistakes (to the nearest whole number):

5. Cities H and K are located on the same line of longitude and the difference in the latitude of these cities is 9°, as shown in the accompanying diagram. If Earth’s radius is 3,954 miles, how far apart are city K and city H along arc HK? Round your answer to the nearest tenth of a mile.

6.

7. Compare the quantity in Column A with the quantity in Column B.

Column A Column B

the arc intercepted by an angle of

in circle X the arc intercepted by an angle of

in circle Y.

(a} The quantity in Column A is greater. (b) The quantity in Column B is greater.

(c)The two quantities are equal. (d) The relationship cannot be determined on the basis of

the information supplied.

Hint. Multiple choice:

Side x is equal to half of

the _____________

A) radius

b) Diameter

c) Area

d) Circumference

unit 12: circle equations · unit 12: circle equations in this unit you must bring the following...

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