# K12MATH010: Geometry

Unit 5: Circles   In this unit, you will deepen your understanding of circles. You will investigate a variety of theorems, with an emphasis on perpendicularity and inscribed angles, helping you to see symmetry in circles and see applications of triangle congruence criteria. You will study additional properties among chords, secants, and tangents as an application of similarity. You will examine circumference and area and further your understanding of these concepts by also learning to calculate arc lengths and areas of sectors of circles. Finally, you will investigate learned theorems and properties of circles using the coordinate plane as an additional resource. Here you will also study the equation of a circle, learning how to identify the equation based on the graphed circle, graph the circle given the equation, and find the intersections between lines and circles or parabolas and between two circles, by solving quadratic equations.[[5]](#_ftn1)

[5]Ibid.

Completing this unit should take you approximately 9 hours.

☐    Subunit 5.1: 3 hours

☐    Subunit 5.2: 3 hours

☐    Subunit 3.3: 3 hours

Unit5 Learning Outcomes
Upon successful completion of this unit, you will be able to:
- Explain and apply theorems about circles. - Find arc lengths and areas of sectors. - Translate between the geometric description and the equation for a conic section. - Use coordinates to prove simple theorems algebraically. - Solve quadratic equations to determine intersections between lines and circles or parabolas and between two circles.

5.1 Properties   In this subunit, you will examine properties and theorems of circles. Here you will continue practicing the logic skills you learned previously, writing proofs related to the circle theorems you study. This section will build on your knowledge of angles and segments and review the basic parts of a circle: chord, tangent, radius, and diameter. The resources in this subunit provide explanatory content as well as practice problems that allow you to receive immediate feedback on your progress and understanding.

5.1.1 Properties of Tangents   - Explanation: CK-12: Geometry: “Parts of Circles & Tangent Lines” Link: CK-12: Geometry: “Parts of Circles & Tangent Lines” (HTML)

Instructions: Please click on the link above, and scroll down to read and take notes on the sections titled “Tangent Lines,” “Tangents and Radii,” and “Tangent Segments.” Complete the examples. Answers follow each example. Upon reading this section, you should be able to explain several properties and theorems of tangents and use them to solve problems.

Completing this activity should take approximately 15 minutes.

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[CCSS.Math.Content.HSG-C.A.2](http://www.corestandards.org/Math/Content/HSG/C/A/2)
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[CCSS.ELA-Literacy.RST.9-10.3](http://www.corestandards.org/ELA-Literacy/RST/9-10/3)
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[CCSS.ELA-Literacy.WHST.9-10.1e](http://www.corestandards.org/ELA-Literacy/WHST/9-10/1/e)

displayed on the webpage above.
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5.1.2 Chords   - Explanation: CK-12: Geometry: “Properties of Chords” Link: CK-12: Geometry: “Properties of Chords” (HTML)

Instructions: Please click on the link above. Read, take notes, and complete the examples as you go. Answers follow each example. Upon reading this section, you should be able to explain several properties and theorems of chords.

Completing this activity should take approximately 15 minutes.

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[CCSS.Math.Content.HSG-C.A.2](http://www.corestandards.org/Math/Content/HSG/C/A/2)
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[CCSS.ELA-Literacy.RST.9-10.3](http://www.corestandards.org/ELA-Literacy/RST/9-10/3)
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[CCSS.ELA-Literacy.WHST.9-10.1e](http://www.corestandards.org/ELA-Literacy/WHST/9-10/1/e)

displayed on the webpage above.
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5.1.3 Arcs and Angles   - Explanation: CK-12: Geometry: “Inscribed Angles” Link: CK-12: Geometry: “Inscribed Angles” (HTML)

Instructions: Please click on the link above. Read, take notes, and complete the examples as you go. Answers follow each example. Upon reading this section, you should be able to explain several properties and theorems of inscribed angles and arcs.

Completing this activity should take approximately 15 minutes.

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[CCSS.Math.Content.HSG-C.A.2](http://www.corestandards.org/Math/Content/HSG/C/A/2)
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[CCSS.ELA-Literacy.RST.9-10.3](http://www.corestandards.org/ELA-Literacy/RST/9-10/3)
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[CCSS.ELA-Literacy.WHST.9-10.1e](http://www.corestandards.org/ELA-Literacy/WHST/9-10/1/e)

displayed on the webpage above.
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• Did I Get This? Activity: Khan Academy’s “Circles and Arcs” Link: Khan Academy’s “Circles and Arcs” (HTML)

Instructions: Please click on the link above. This page provides a series of practice problems that allow you test your knowledge of arcs in circles. Each question has explanations worked out step-by-step if you need hints along the way. The program monitors your success, and when you have correctly completed about six questions consecutively, it will tell you that you are ready to move on to a new skill.

Completing this activity should take approximately 30 minutes.

●      CCSS.Math.Content.HSG-C.A.2

5.1.4 Angles of Chords, Secants, and Tangents   - Explanation: CK-12: Geometry: “Angles of Chords, Secants, and Tangents” Link: CK-12: Geometry: “Angles of Chords, Secants, and Tangents” (HTML)

Instructions: Please click on the link above. Read, take notes, and complete the examples as you go. Answers follow each example. Upon reading this section, you should be able to explain several properties and theorems of inscribed angles and arcs.

Completing this activity should take approximately 15 minutes.

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[CCSS.Math.Content.HSG-C.A.2](http://www.corestandards.org/Math/Content/HSG/C/A/2)
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[CCSS.ELA-Literacy.RST.9-10.3](http://www.corestandards.org/ELA-Literacy/RST/9-10/3)
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[CCSS.ELA-Literacy.WHST.9-10.1e](http://www.corestandards.org/ELA-Literacy/WHST/9-10/1/e)

displayed on the webpage above.
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• Did I Get This? Activity: Khan Academy’s “Inscribed Angles 1” Link: Khan Academy’s “Inscribed Angles 1” (HTML)

Instructions: Please click on the link above. This page provides a series of practice problems that allow you test your understanding of the relationship between inscribed angles and central angles. Each question has explanations worked out step-by-step if you need hints along the way. The program monitors your success, and when you have correctly completed about six questions consecutively, it will tell you that you are ready to move on to a new skill.

Completing this activity should take approximately 30 minutes.

●      CCSS.Math.Content.HSG-C.A.2

• Did I Get This? Activity: MoodleShare: “Guided Practice for Secants, Tangents, and Angle Measurements” Link: MoodleShare: “Guided Practice for Secants, Tangents, and Angle Measurements” (HTML)

Instructions: Please click on the link above, and then click on the link on the page that appears. This webpage provides a series of practice problems that you can answer and check online.

Completing this activity should take approximately 15 minutes.

●      CCSS.Math.Content.HSG-C.A.2

• Activity: Wikispaces: everman164’s “Inscribed Angles Practice” Link: Wikispaces: everman164’s “Inscribed Angles Practice” (DOC)

Instructions: Please click on the link above, and then click on the link next to “Filename” in order to download the practice problems. This document provides a series of questions for solving for unknown values by using your knowledge of inscribed angles.

Completing this activity should take approximately 30 minutes.

●      CCSS.Math.Content.HSG-C.A.2

5.1.5 Segments of Chords, Secants, and Tangents   - Explanation: CK-12: Geometry: “Segments of Chords, Secants, and Tangents” Link: CK-12: Geometry: “Segments of Chords, Secants, and Tangents” (HTML)

Instructions: Please click on the link above. Read, take notes, and complete the examples as you go. Answers follow each example. Upon reading this section, you should be able to explain several properties and theorems of chords, secants, and tangents.

Completing this activity should take approximately 15 minutes.

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[CCSS.Math.Content.HSG-C.A.2](http://www.corestandards.org/Math/Content/HSG/C/A/2)
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[CCSS.ELA-Literacy.RST.9-10.3](http://www.corestandards.org/ELA-Literacy/RST/9-10/3)
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[CCSS.ELA-Literacy.WHST.9-10.1e](http://www.corestandards.org/ELA-Literacy/WHST/9-10/1/e)

displayed on the webpage above.
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5.2 Measurement   You have already seen area and perimeter, but with circles, things get a little bit more complicated. This is another opportunity to build up comfort with curved figures that you will see again when studying trigonometric functions and area under the curve in higher level math classes. Additionally, you will find the circumference and parts of the circumference using what you learn. The resources in this subunit provide explanatory content as well as practice problems that allow you to receive immediate feedback on your progress and understanding.

5.2.1 Circumference and Arc Length   - Explanation: CK-12: Geometry: “Circumference and Arc Length” Link: CK-12: Geometry: “Circumference and Arc Length” (HTML)

Instructions: Please click on the link above. Read, take notes, and complete the examples as you go. Answers follow each example. Upon reading this section, you should be able to calculate circumference, define the length of an arc, and find the arc length.

Completing this activity should take approximately 15 minutes.

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[CCSS.Math.Content.HSG-C.B.5](http://www.corestandards.org/Math/Content/HSG/C/B/5)
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[CCSS.ELA-Literacy.RST.9-10.3](http://www.corestandards.org/ELA-Literacy/RST/9-10/3)
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[CCSS.ELA-Literacy.WHST.9-10.1e](http://www.corestandards.org/ELA-Literacy/WHST/9-10/1/e)

displayed on the webpage above.
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Instructions: Please click on the link above. This page provides a series of practice problems that allow you to calculate the radius, diameter, and circumference in relationship to the others. Each question has explanations worked out step-by-step if you need hints along the way. The program monitors your success, and when you have correctly completed about six questions consecutively, it will tell you that you are ready to move on to a new skill.

Completing this activity should take approximately 30 minutes.

●      CCSS.Math.Content.HSG-C.B.5

5.2.2 Areas of Circles and Sectors   - Explanation: CK-12: Geometry: “Areas of Circles and Sectors” Link: CK-12: Geometry: “Areas of Circles and Sectors” (HTML)

Instructions: Please click on the link above. Read, take notes, and complete the examples as you go. Answers follow each example. Upon reading this section, you should be able to calculate the area of a circle, find the radius or diameter when given the area of the circle, and find the area of sectors of a circle.

Completing this activity should take approximately 15 minutes.

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[CCSS.Math.Content.HSG-C.B.5](http://www.corestandards.org/Math/Content/HSG/C/B/5)
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[CCSS.ELA-Literacy.RST.9-10.3](http://www.corestandards.org/ELA-Literacy/RST/9-10/3)
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[CCSS.ELA-Literacy.WHST.9-10.1e](http://www.corestandards.org/ELA-Literacy/WHST/9-10/1/e)

displayed on the webpage above.
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• Did I Get This? Activity: Khan Academy’s “Areas of Circles and Sectors” Link: Khan Academy’s “Areas of Circles and Sectors” (HTML)

Instructions: Please click on the link above. This page provides a series of practice problems that allow you to calculate the areas and sectors of circles. Each question has explanations worked out step-by-step if you need hints along the way. The program monitors your success, and when you have correctly completed about six questions consecutively, it will tell you that you are ready to move on to a new skill.

Completing this activity should take approximately 30 minutes.

●      CCSS.Math.Content.HSG-C.B.5

Instructions: Please click on the link above. This page provides a series of practice problems that allow you to test what you have learned about circles. Each question has explanations worked out step-by-step if you need hints along the way. The program monitors your success, and when you have correctly completed about six questions consecutively, it will tell you that you are ready to move on to a new skill.

Completing this activity should take approximately 30 minutes.

●      CCSS.Math.Content.HSG-C.B.5

• Checkpoint: Mathematics Assessment Project: “Geometry Problems: Circles and Triangles” Link: Mathematics Assessment Project: “Geometry Problems: Circles and Triangles” (HTML)

Instructions: Please click on the link above, and scroll down to the section of the lesson entitled “Resources.” Download the PDF file underneath “Lesson (complete).” You will find the assessment on page S-1, titled “Circles and Triangles.” Complete the questions to the best of your ability. These questions incorporate many of the skills you have seen thus far in the course. You will need to draw on your understanding of circle and triangles, as well as area and proportions. When you are finished, use the solutions on pages T-9 through T-11 to check your understanding. For an additional challenge, scroll down to the sample responses. Questions are provided to guide you in the analysis of the work and in making improvements to each sample.

Completing this activity should take approximately 45 minutes. The extension will add another 30 minutes.

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●      CCSS.Math.Content.HSG-SRT.C.8
●      CCSS.Math.Content.HSG-MG.A.1
●      CCSS.Math.Content.HSG-C.A.2
●      CCSS.ELA-Literacy.WHST.9-10.1d
●      CCSS.ELA-Literacy.WHST.9-10.1e

5.3 Circles on the Coordinate Plane   This subunit provides one more way for you to understand circles. You will use what you learned about geometry on the coordinate plane to investigate circles. The resources in this subunit provide explanatory content as well as practice problems that allow you to receive immediate feedback on your progress and understanding.

• Explanation: CK-12: Geometry: “Extension: Writing and Graphing the Equations of Circles” Link: CK-12: Geometry: “Extension: Writing and Graphing the Equations of Circles” (HTML)

Instructions: Please click on the link above. Read, take notes, and complete the examples as you go. Answers follow each example. Upon reading this section, you should be able to graph a circle, write the equation of a circle given the key information, and find the radius and center given the equation of the circle.

Completing this activity should take approximately 15 minutes.

●      CCSS.Math.Content.HSG-GPE.A.1
●      CCSS.ELA-Literacy.RST.9-10.3
●      CCSS.ELA-Literacy.WHST.9-10.1e

• Web Media: Open Math Reference: “Basic Equation of a Circle” Link: Open Math Reference: “Basic Equation of a Circle” (Flash)

Instructions: Please click on the link above, and then click on the point on the radius of the circle. Move it using your mouse and notice how the equation of the circle changes with the radius. Read the content below the coordinate plane, which reviews what you read from the CK-12 textbook, Geometry. When you reach “Things to Try,” repeat the activity until you can complete it successfully three times.

Completing this activity should take approximately 45 minutes.

●      CCSS.Math.Content.HSG-GPE.A.1

• Web Media: Open Math Reference: “General Form of Equation of a Circle” Link: Open Math Reference: “General Form of Equation of a Circle” (Flash)

Instructions: Please click on the link above, and then click on the center of the circle. Move the center of the circle using your mouse, and notice how the equation of the circle changes as the circle’s center changes. Read the content below the coordinate plane, which reviews what you read from the CK-12 textbook, Geometry. When you reach “Things To Try,” repeat the activity until you can complete it successfully three times.

Completing this activity should take approximately 45 minutes.

●      CCSS.Math.Content.HSG-GPE.A.1

• Did I Get This? Activity: NASA: “The InSight Mission—The Work Area In Front of the Lander” Link: NASA: “The InSight Mission—The Work Area In Front of the Lander” (PDF)

Instructions: Please click on the link above, and the scroll down to problem 2. Answer the question using coordinate geometry, and then scroll down to check your answer.

Completing this activity should take approximately 30 minutes.

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●      CCSS.ELA-Literacy.RST.9-10.3
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• Did I Get This? Activity: Illustrative Mathematics: “A-REI: The Circle and the Line” Link: Illustrative Mathematics: “A-REI: The Circle and the Line” (HTML)

Instructions: Please click on the link above, and read and complete the task described. When you are done, you should scroll down and read through the detailed solution provided for this problem, to ensure that you correctly solved the problem and that you could repeat this process for similar problems.

Completing this activity should take approximately 15 minutes.

●      CCSS.Math.Content.HSG-GPE.A.1
●      CCSS.ELA-Literacy.RST.9-10.3