Geometry Common Core Circles Packet Part One - White ...

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Geometry Common Core

Circles Packet

Part One

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Name: _______________________________________ 10-1 Notes

Geometry Pd. ____ Date: ______

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

Warm - Up

Use the Distance Formula to find the distance, in simplest radical form, between the pair of points.

A(6, 2) and D(–2, –2)

Flashback! What do we call the segment from a to P? b to P? How do they relate? Why/ how do we know this?

What is the name of the circle?

Since all points are equidistant, let's find the distance they all are from the center point!

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Example 1

Example 2

Big Take Away!

Look at the two problems we just did...

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 its represented in the equation of a circle?

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?

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

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

Example 5 - Graphing Circles Graph the circle whose equation is: x2 + (y - 2)2 = 25

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10-1 Practice

1) Given the equation of a circle (x + 3)2 + (y - 6)2 = 24, what are the coordinates of the center of the circle

and the radius (in simplest radical form)?

2) Write the equation of the circle centered at (-2, 3) with a radius of 13.

3) What is the equation for the circle shown in the graph to the right?

(Hint: First identify the center and radius)

4) Graph the circle with an equation of (x + 4)2 + (y – 2) 2 = 16.

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5) Casa D'italia is located in downtown Harrison at the point (4, -1) on the coordinate plane. The delivery

service extends 8 miles. Write the equation of the circle which represents the outer edge of the delivery

service.

6) Identify the center and radius of the following circle: 3x2 + 3y2 = 75

7) Given the equation of a circle (x - 8)2 + (y + 1)2 = 4, what are the coordinates of the center and the radius?

8) Write the equation of the circle centered at (0, -5) with a radius of 11.

9) Pizza 2000 is located in downtown Harrison at the point (-7, -3) on the coordinate plane. The delivery

service extends 12 miles. Write the equation of the circle which represents the outer edge of the delivery

service.

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10) Write the equation for circle K, shown in the graph below.

11) Graph and write an equation of the circle whose diameter has endpoints A(1, 3) and B(-5, 3)

12) Error Analysis: Sydney was given the following equation of a circle to graph. Describe her mistake(s), and explain

why she may have made it/them.

(𝑥 − 1)2 + (𝑦 + 2)2 = 4

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Challenge SUMMARY

Exit Ticket

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Name Date

Geometry Pd. _______ 10-1 HW

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 √6. 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√7 miles. Write the equation that will represent the outer edge of the broadcasting range.

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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

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Name: ________________________________________________ 10-2 Notes

Geometry Pd. _____ Date: ________

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

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. 𝑥2 + 12𝑥 + 36

b. 𝑥2 − 8𝑥 + 16

c. 𝑦2 + 14𝑦 + 49

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 (______)2 .

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 (______)2 .

Why are these types of trinomials necessary?

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Creating perfect square trinomials within an equation:

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

Then write in factored form.

𝑥2 + 6𝑥 + 1 = 1

You try one!

𝑦2 − 8𝑦 + 24 = 0

Class discussion

What is different here in this equation?

𝑥2 + 𝑦2 − 4𝑥 − 6𝑦 + 8 = 0

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

Let’s do it!

𝑥2 + 𝑦2 − 4𝑥 − 6𝑦 + 8 = 0

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.

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You try!

x2 + y2 – 6x + 4y – 3 = 0

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.

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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?

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Name: ________________________________________________ 10-2 HW

Geometry Pd. _____ Date: ___

1.

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

a. x2 – 6x + y2 + 4y = 12

b.

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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.

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Name: ______________________________________ 10 - 3 Lesson

Geometry Pd. __ Date: _______

With your squad Use the equation shown right to answer #1-3: 𝑥2 + 𝑦2 − 4𝑥 − 6𝑦 + 8 = 0

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?

The equation shown above IS a circle!

We call this form the 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 (x − h)2 + (y − k)2 = r2 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.

Back to your squad!

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

𝑥2 + 𝑦2 − 4𝑥 − 6𝑦 − 3 = 0

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

x2 + 2x + y2 - 4y = 3

What method did you use?

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Example 3) The Equations of a circle is 𝑥2 + 𝑦2 + 6𝑦 = 7 . What are the coordinates of the center and the length of

the radius of the circle?

Example 4) Use the following general form for a-c: 𝑥2 + 𝑦2 − 8𝑥 + 4𝑦 − 16 = 0

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

b) Identify the center and radius of the circle

c) Graph the circle

What’s different here?

Do I need to complete the

square twice?

Remember! Circles of the

form 𝑥2 + 𝑦2 = 𝑟2 have a

center at the origin!

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Example 5) The point (3,4) is on a circle whose center is (1,4).

Write the center-radius form of the circle

Example 6) Show that (𝑥 − 2)(𝑥 − 6) + (𝑦 − 5)(𝑦 + 11) = 0 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 of the equation to the right.

Careful! spicy

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Name: ________________________________________________ 10-3 HW


1. Each equation below represents a circle. Place its equation in standard form and then state the coordinates of its

center and the length of its radius.

(a) 2 28 2 8x x y y (b) 2 26 20 12x x y y

(c) 2 216 10 88 0x x y y (d)

2 2 8 84y x x

(e) 2 23 42 3 12 33x x y y (f)

2 22 2 20 24 166x y x y

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2. The radius of a circle whose equation is 2 28 6 13x x y y is closest to

(1) 3.6 (3) 5.3

(2) 4.6 (4) 6.2

3. Which of the following would be the maximum y-value on the circle whose equation is shown below?

(1) 5y

(2) 2y

(3) 12y

(4) 22y

4. Which of the following is the length of the diameter of the circle whose equation is shown below?

(1) 5

(2) 8

(3) 10

(4) 16

2 210 4 71x x y y

2 2 16 39 0x y x

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Name: ________________________________________________ 10-4 Notes


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.

𝑦 = 𝑥 + 7 𝑥2 + 𝑦2 = 49

Line Circle

How can we be sure of our solutions?

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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

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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!

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5. Solve the following system of equations.

6. Solve the following system of equations

x + y = -2

x2 + y2 -4x – 2y = 20

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Name________________________________ Date__________ Geometry Period________ Hw 10-4

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.

(𝑥 + 1)2 + (𝑦 + 1)2 = 16

𝑥 + 𝑦 = 2

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!

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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

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Name: ___________________________________________ 10-5 Notes

Geometry Pd. _____ Date: ____

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:

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?

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Try it!

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

Example 2:

The circle from example 1 was split up! What is the area of the shaded region in terms of 𝜋?

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? _______________________

What is the area of the shaded region? _______________________

The circle from ex 1, was split up again! What is the area of the shaded region in terms of 𝜋?

But first!!!!






The circle from ex 1, was split up again! What is the area of the shaded region in terms of 𝜋?




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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?

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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?

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Name: ___________________________________________ 10-5 HW

Geometry Pd. _____ Date: ____

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 𝜋.

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

Don’t forget

your learning

goals!

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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.

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Name: _________________________________ 11-7 Notes Geometry Per __________ Date:_________

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)

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Example 1)

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

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

My thinking:

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

How many degrees is the marked arc? _____________________________

How many degrees is the full circle? _______________________________

What fraction of the circle is the arc of the shaded region? _____________

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

Arc Length of a Circle:

Let’s try one!

The length of PA is 8 cm 𝑚∡𝐴𝑃𝐵 = 60 °. What is the length of the minor arc 𝐴�̂� in terms of π?

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Practice

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

nearest tenth of a meter?

2. The length of 𝑋�̂� is 4.19 in and 𝑚∡𝑋𝑍𝑌 = 40 °. 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? Round to the nearest tenth.

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4. Find the length of minor arc 𝐴�̂� to the nearest tenth:

5.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 𝜋?

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Name: _________________________________ 10-6 Homework

Geometry Per __________ Date:_________

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.

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Name: _________________________________ 10-7 Homework

Geometry Per __________ Date:_________

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

Warm-Up: Answer following question.

1) What is the length of the minor arc below to the nearest whole number?

Check In

1. Write the formula for the circumference of a circle:

2. What is the name of the major arc?

Minor Arc?

3. 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” _____

C =

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Degrees Radians

*We already know that the number of degrees in a full circle is _________________.

*We already know that the number of degrees in a semi-circle is ________________.

What is a radian? Let’s Look at 3 Radians:

*How many radians would be in a semi-circle?

________________ radians

Apply it! Using this relationship, how might we be able to figure out the number

of radians that equate to 60o? LEAVE IN TERMS OF PI!

You try: Convert 45 degrees to radians! LEAVE IN TERMS OF PI!

Convert the following to degrees: 𝜋

2 radians

**Note: Always leave

radians in terms of 𝝅, and as

a reduced fraction, unless

otherwise specified.

*SET UP PROPORTION:*

𝑫

𝑹=

𝟏𝟖𝟎

𝝅

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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

Examples

1) In a circle, the length of a radius is 4cm. Find the length of an arc intercepted by a central

angle whose measure is 1.5 radians.

2) A 20" pendulum swings through an angle of 1.5 radians. What is the distance covered by the tip of the

pendulum?

3) Find the length of the arc when Ө = 55 and the radius is 1.25.

Watch out! r is radius NOT

radians!

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You try it!

4) A circle has radius 1.7 inches. Find the length of an arc intercepted by a central angle of 2

radians.

5) A ball rolls in a circular path that has a radius of 5 inches, as shown in the accompanying

diagram. If the ball rolls through an angle of 2 radians, find the distance traveled by the ball.

6) 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 subtended arc is 54°?

Express your answer to the nearest hundredth of an inch.

2 radians

5

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Exit Ticket

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Name: ______________________________________ 10-7 Homework

Geometry Per ______ Date: _______

*Don’t forget about your looking forward!

1. Convert the following:

a) 3𝜋

2 radians b) 240°

2. What is the value of the arc length, s, in the circle below in terms of 𝜋? *Note this angle is in radians

3. An arc of a circle that is 6 centimeters in length intercepts a central angle of 1.5 radians. Find the number of centimeters in the radius of the circle.

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4. 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.

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

whole number):

7) 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 many miles north of city K is city H along arc

HK? Round your answer to the nearest tenth of a mile.

What method did you use?

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5. Regents Style Question about Arc Length

Hint. Multiple choice:

Side x is equal to half of

the _____________

A) radius

b) Diameter

c) Area

d) Circumference

46

Name: _________________________________________________________ Unit 10 – Part One Review

Geometry Pd. ______ Date: __________

Circle Summary Sheet

Big Ideas Topic Overview

10-1 Center Radius Form, for circle centered at (h,k) with radius, r (𝑥 − ℎ)2 + (𝑦 − 𝑘)2 = 𝑟2

10-2 Completing the square for center-radius form 1. Move loose numbers to one side 2. Group x’s and y’s 3. Divide middle term by 2 and square it – ADD TO BOTH SIDES! 4. put factors into Squared Form ( )2 ( remember the number will be half of the middle term) 5. You’re in center-radius form!!!

10-3 We complete the square twice to put general form equations of circles into Center-Radius form, then graph!

Recognize a circle by finding an 𝑥2 and 𝑦2

10-4 Systems with Circles Any point of intersection is a solution to the system – solve graphically- BRING COMPASS

10-5 Area of a Sector

𝐴𝑟𝑒𝑎 𝑜𝑓 𝑎 𝑆𝑒𝑐𝑡𝑜𝑟 = 𝐴𝑟𝑒𝑎 𝑜𝑓 𝑎 𝐶𝑖𝑟𝑐𝑙𝑒 (𝑎𝑛𝑔𝑙𝑒 𝑚𝑒𝑎𝑠𝑢𝑟𝑒 𝑜𝑓 𝑠𝑒𝑐𝑡𝑜𝑟

360)

Area of a circle = 𝜋𝑟2

10-6 Arc Length of a Sector (In degrees)

𝐴𝑟𝑐 𝐿𝑒𝑛𝑔𝑡ℎ = 𝐶𝑖𝑟𝑐𝑢𝑚𝑓𝑒𝑟𝑒𝑛𝑐𝑒 (𝑎𝑛𝑔𝑙𝑒 𝑚𝑒𝑎𝑠𝑢𝑟𝑒 𝑜𝑓 𝑠𝑒𝑐𝑡𝑜𝑟

360)

Circumference of a circle = 𝜋𝑑

10-7 Solving for arc length IN RADIANS

s = 𝜃𝑟 where: s = arc length; r = radius; 𝜃 = central angle Radians – unit of angle measure An angle is 1 radian when the length of the arc of the circle is equal to the radius

47

Station 1: Circle Equations

1. Write the equation of a circle with a radius of √5 units and a center (3,-2).

2. Identify the center and radius of the following circle. Leave answers in simplest radical form.

𝑥2 + (𝑦 + 5)2 = 50

3. Use the following equation for parts a-c

a) Write the equation in center-radius form

b) Identify the center and radius of the circle

center: __________ radius: ___________

c) Graph the circle →

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4. Write the equation of the circle graphed below:

5. Write the equation of a circle whose center is (2,1) and passes through the point (2,-3)

6. Graph the following circle: 7𝑥2 + 7𝑦2 = 448

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7. Graph the following circle:

x2 – 2x + y2 + 8y – 8 = 0

Station 2: Systems

1. Solve the following System of equations graphically:

y = x2 +2x – 3

y + x = 1

2. How many solutions does this system have?

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y = x2 - x - 6

y = 2x -8

What quadrant does the solution (s) fall in?

3. Solve the following system of equations.

State all solutions.

(𝑥 − 4)2 + (𝑦 + 2)2 = 9

𝑦 = 1

4. Solve the following system of equations. State all solutions.

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2𝑥2 + 2𝑦2 = 2

𝑥2 + (𝑦 − 3)2 = 4

Station 3: Area of a Sector

1. Segment NM is 3.5 in. Find the area of the shaded region to the nearest whole number

2. The radius of the circle shown below is 8 in. Find the area of the shaded region to the nearest whole number:

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3. The area of sector FHE is 156.38 𝑦𝑑2

Station 4: Arc Length in degrees and/or radians

Convert the following:

1. 300° = _____________ radians

2. ________________ degrees = 𝜋

12

3. a) The length of minor arc SQ is 3.82 in. Find the circumference of circle R to the nearest inch.

b) What is the radius of circle R to the nearest tenth?( use your final answer from part A)

Round to the nearest tenth.

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4) A central angle of a circular garden measures 2.5 radians and intercepts an arc of 20 feet. What is the radius of the

garden?

1) 8 ft

2) 50 ft

3) 100 ft

4) 125 ft

5) In a circle, a central angle intercepts an arc of 12 centimeters. If the radius of the circle is 6 centimeters, find the

number of radians in the measure of the central angle.

6) In a circle whose radius is 10 ft., what is the length of the arc intercepted by a central angle of 4 radians

7) The accompanying diagram shows the path of a cart traveling on a circular track of radius 2.40 meters. The cart starts

at point A and stops at point B, moving in a counterclockwise direction. What is the length of minor arc AB, over which

the cart traveled, to the nearest tenth of a meter?

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Self-Assess for Success!

Fill in the following chart for each topic by placing a check mark in the box

that describes your knowledge of each topic.

** Be honest! It’s just you looking at this! **

Topic This is easy

for me

This is o.k.

for me

This is

really

difficult for

me

1. Graphing Circles

2. Writing Circle

Equations

3. Completing the

square to get equation

of circle in center-

radius form

4. Graphing Parabolas

5. Solving Systems

Graphically

6. Area of a Sector

7. Arc Length Given

Degrees

8. Arc Length Given

Radians

9. Converting degrees

to radians

#1-3 Start at

station 1

#4 – 5 Start at

station 2

#6 Start at

Station 3

#7-9 Start at

station 4

Geometry Common Core Circles Packet Part One - White ...

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