Circles

Lesson 9

Math

Unit 7

10th Grade

Lesson 9 of 14

Objective


Construct tangent lines to a circle to define and describe the circumscribed angle. 

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.
  • G.C.A.4 — Construct a tangent line from a point outside a given circle to the circle.

Foundational Standards

  • G.CO.D.13

Criteria for Success


  1. Using the property that a radius is perpendicular to the tangent line at the point of tangency, identify the constructions necessary to construct a tangent line to a point on the circle.
  2. Define a circumscribed angle as an angle formed by two tangent lines that intersect a point outside the circle.
  3. Describe the relationship between a circumscribed angle and the central angle that meet at the two points of tangency as supplementary. See Unit 7 Glossary for a visual.
  4. Use properties of tangent lines to solve problems.

Tips for Teachers


  • This lesson addressed standard G-C.4 (+), which is represented in its fullest in the problem set guidance. Anchor Problem #1 asks students to construct a tangent line given a point on the circle, whereas G-C.4(+) states, “Construct a tangent line from a point outside a given circle to the circle.”
  • The following tools will be needed for this lesson: compass and straightedge. 
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Anchor Problems


Problem 1

Construct a tangent line to circle $$A$$ at point $$B$$.

Guiding Questions

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References

Math Open Reference Tangent to a Circle at a Point

Tangent to a Circle at a Point by John D. Page is made available on Math Open Reference. © 2011 Copyright Math Open Reference. All rights reserved. Accessed Sept. 19, 2018, 2:04 p.m..

Problem 2

Below is circle $$A$$ with tangent lines $$\overleftrightarrow{BD}$$ and $$\overleftrightarrow{CD}$$.

What is the relationship between the central angle and the circumscribed angle?

Guiding Questions

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References

GeoGebra Geometry - 8.9 AP2

Geometry - 8.9 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, 11:49 a.m..

Target Task


  1. Draw a circle tangent to both rays of this angle.

  1. Let $$B$$ and $$C$$ be the points of tangency of your circle. Find the measures of $$\angle ABC$$ and $$\angle ACB$$. Explain how you determined your answer.
  2. Let $$P$$ be the center of your circle. Find the measures of the angles in $$\triangle APB$$.

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 8

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Lesson 10

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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