Circles

Lesson 4

Math

Unit 7

10th Grade

Lesson 4 of 14

Objective


Define a chord to derive the Chord Central Angles Conjecture and Thales’ Theorem. 

Common Core Standards


Core Standards

  • G.C.A.2 — Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle.

Foundational Standards

  • 7.G.B.4
  • 7.G.B.5

Criteria for Success


  1. Define, describe, and compare the features of a circle, including chord, diameter, tangent, radius, arc, and central angle.
  2. Derive theorems about chords and diameters in a circle and their converses such as:
    1. If two chords are congruent, then their central angles are equal in measure. (Chord Central Angles Conjecture) See Unit 7 Glossary for a visual.
    2. If two chords define central angles equal in measure, then they are congruent. (Chord Central Angles Conjecture) See Unit 7 Glossary for a visual.
    3. If a triangle is inscribed in a circle where one of the sides is a diameter, then the triangle is a right triangle. (Thales’ Theorem) See Unit 7 Glossary for a visual.
  3. Apply the Chord Central Angles Conjecture and Thales’ Theorem to find missing measures and establish other relationships and conjectures in circles. 

Tips for Teachers


  • The GeoGebra tool Geometry - 8.4 AP1 is helpful to show students how a chord can also be a diameter when it goes through the center. By moving point $$C$$ or point $$D$$ in the file, you can show the arc changing length as well as move the chord so that it goes through the center to become a diameter. If you unhighlight the tangent as you do this, it will make the definitions easier to see as well. 
  • This lesson defines an arc, but the students will not measure arcs until Lesson 5 (measured by the degree of turn) and Lesson 11 (measured in radians as the length of the circumference). 
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Anchor Problems


Problem 1

Based on the definition of each feature of a circle below, label the circle with the appropriate vocabulary words.

Arc: Part of any curve
Central angle: An angle formed by two radii with the vertex as the center of the circle
Chord: A line segment containing two points on a curve
Diameter: A line segment connecting two points on a curve that passes through the center of the circle
Radius: A line segment that connects the center of a circle and a point on the circle 
Tangent: A line that intersect a circle at one and only one point

 

Guiding Questions

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References

GeoGebra Geometry - 8.4 AP1

Geometry - 8.4 AP1 by Match Fishtank is made available by GeoGebra under the CC BY-NC-SA 3.0 license. Copyright © International GeoGebra Institute, 2013. Accessed June 13, 2017, 11:02 a.m..

Problem 2

Below is circle $$A$$  with points $$E$$, $$C$$, $$D$$, and $$B$$ on the circle. $$\angle EAC \cong \angle DAB$$.

What is the relationship between $$\overline{BD}$$ and $$\overline {EC}$$?

Guiding Questions

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References

GeoGebra Geometry - 8.4 AP2

Geometry - 8.4 AP2 by Match Fishtank is made available by GeoGebra under the CC BY-NC-SA 3.0 license. Copyright © International GeoGebra Institute, 2013. Accessed June 13, 2017, 11:04 a.m..

Problem 3

Given diameter $$\overline{CB}$$, chords $$\overline{DB}$$ and $$\overline{CD}$$, and radii $$\overline{AC}$$$$\overline{AD}$$, and $$\overline{AB}$$, find the measure of $$\angle CDB$$.

Guiding Questions

Create a free account or sign in to access the Guiding Questions for this Anchor Problem.

References

GeoGebra Geometry - 8.4 AP3

Geometry - 8.4 AP3 by Match Fishtank is made available by GeoGebra under the CC BY-NC-SA 3.0 license. Copyright © International GeoGebra Institute, 2013. Accessed June 13, 2017, 11:07 a.m..

Target Task


Circle $$A$$ is shown below.

  1. Draw two diameters of the circle.
  2. Identify the shape defined by the end points of the two diameters.
  3. Explain why this shape is always the result. 

References

EngageNY Mathematics Geometry > Module 5 > Topic A > Lesson 1Exit Ticket

Geometry > Module 5 > Topic A > Lesson 1 of the New York State Common Core Mathematics Curriculum from EngageNY and Great Minds. © 2015 Great Minds. Licensed by EngageNY of the New York State Education Department under the CC BY-NC-SA 3.0 US license. Accessed Dec. 2, 2016, 5:15 p.m..

Additional Practice


The following resources include problems and activities aligned to the objective of the lesson that can be used for additional practice or to create your own problem set.

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Lesson 3

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Lesson 5

Lesson Map

A7CB09C2-D12F-4F55-80DB-37298FF0A765

Topic A: Equations of Circles

Topic B: Angle and Segment Relationships in Inscribed and Circumscribed Figures

Topic C: Arc Length, Radians, and Sector Area

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