Lesson 8 | Circles | 10th Grade Mathematics

Objective

Define and determine properties of tangents and secants of circles to solve problems with inscribed and circumscribed triangles.

Common Core Standards

Core Standards

The core standards covered in this lesson

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.

Circles

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.3 — Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle.

Circles

G.C.A.3 — Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle.

Foundational Standards

The foundational standards covered in this lesson

G.CO.B.8

Congruence

G.CO.B.8 — Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions.
G.CO.D.13

Congruence

G.CO.D.13 — Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle.

Criteria for Success

The essential concepts students need to demonstrate or understand to achieve the lesson objective

Define and identify tangents, secants, and points of tangency in a circle.
Prove that a line is tangent to a circle if the radius drawn to the point of tangency is perpendicular to the line. See Unit 7 Glossary for a visual.
Describe that two tangent lines that intersect at a point outside the circle have congruent line segments formed from their point of intersection to their point of tangency. See Unit 7 Glossary for a visual.
Describe how knowing the relationship between tangents and radii of circles can help to solve problems involving inscribed and circumscribed triangles.

Tips for Teachers

Suggestions for teachers to help them teach this lesson

This lesson will prepare students to access G-C.4(+) standard by introducing them to tangents and secants of circles. In the next lesson, students will be constructing a tangent line.
This lesson connects to Unit 1, Lesson 15 with inscribed and circumscribed triangles. Therefore, CFS4 is addressed mostly in the problem set guidance.

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Anchor Problems

Problems designed to teach key points of the lesson and guiding questions to help draw out student understanding

Problem 1

Compare the two diagrams below. How are they similar? How are they different?

Guiding Questions

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References

GeoGebra Geometry - 8.8 AP1

Problem 2

Below is circle $$A$$ and circle $$C$$. $$\overline{CG}$$ is a diameter of circle $$A$$.$$\overleftrightarrow{EG}$$ is tangent to circle $$C$$ at point $$E$$. $$\overleftrightarrow{FG}$$ is tangent to circle $$C$$ at point $$F$$.

What are the angle measures between the tangent lines and the radii of circle $$C$$ with endpoints of $$E$$ and $$F$$?
What is the relationship between $$\overline{EG}$$ and $$\overline{FG}$$?

Guiding Questions

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References

GeoGebra Geometry - 8.8 AP2

Modified by Fishtank Learning, Inc.

Illustrative Mathematics Tangent to a Circle from a Point

Tangent to a Circle from a Point, accessed on June 13, 2017, 11:52 a.m., is licensed by Illustrative Mathematics under either the CC BY 4.0 or CC BY-NC-SA 4.0. For further information, contact Illustrative Mathematics.

Modified by Fishtank Learning, Inc.

Target Task

A task that represents the peak thinking of the lesson - mastery will indicate whether or not objective was achieved

The radius of circle $$A$$ shown below has a value of 2.5 cm.

What is the length of $$\overline{AD}$$?

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.

EngageNY Mathematics Geometry > Module 5 > Topic C > Lesson 11
Illustrative Mathematics Inscribing a Circle in a Triangle II — (Also ask how inscribing a circle inside a triangle produces a tangent)
Illustrative Mathematics Inscribing a Circle in a Triangle I — (Also ask, "What line segments have to be the radii of the triangle? How do you know?)
Illustrative Mathematics Tangent Lines and the Radius of a Circle

Lesson 7

Lesson 9

Lesson Map

Topic A: Equations of Circles

Derive the equation of a circle using the Pythagorean Theorem where the center of the circle is at the origin.

G.GPE.A.1 G.GPE.B.4

Given a circle with a center translated from the origin, write the equation of the circle and describe its features.

G.C.A.1 G.CO.A.5 G.GPE.A.1

Write an equation for a circle in standard form by completing the square. Describe the transformations of a circle.

G.CO.A.5 G.GPE.A.1

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

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

G.C.A.2

Describe the relationship between inscribed and central angles in terms of their intercepted arc.

G.C.A.2

Determine the angle and length relationships between intersecting chords.

G.C.A.2

Prove properties of angles in a quadrilateral inscribed in a circle.

G.C.A.3

Define and determine properties of tangents and secants of circles to solve problems with inscribed and circumscribed triangles.

G.C.A.2 G.C.A.3

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

G.C.A.2 G.C.A.4

Use angle and side length relationships with chords, tangents, inscribed angles, and circumscribed angles to solve problems.

G.C.A.2

Topic C: Arc Length, Radians, and Sector Area

Define, describe, and calculate arc length.

G.C.B.5

Describe the proportional relationship between arc length and the radius of a circle. Convert between degrees and radians to write the arc measure in radians.

G.C.B.5

Calculate the sector area of a circle. Identify relationships between sector area, arc angle, and radius.

G.C.B.5

Use sector area of circles to calculate the composite area of figures.

G.C.B.5 N.Q.A.2 N.Q.A.3

Circles

Objective

Common Core Standards

Core Standards

Circles

Circles

Foundational Standards

Congruence

Congruence

Criteria for Success

Tips for Teachers

Anchor Problems

Problem 1

Guiding Questions

References

Problem 2

Guiding Questions

References

Target Task

Additional Practice

Lesson Map

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