Curriculum / Math / 10th Grade / Unit 7: Circles / Lesson 8
Math
Unit 7
10th Grade
Lesson 8 of 14
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Lesson Notes
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Define and determine properties of tangents and secants of circles to solve problems with inscribed and circumscribed triangles.
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.
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.
The foundational standards covered in this lesson
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 — Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle.
The essential concepts students need to demonstrate or understand to achieve the lesson objective
Suggestions for teachers to help them teach this lesson
Unlock features to optimize your prep time, plan engaging lessons, and monitor student progress.
Problems designed to teach key points of the lesson and guiding questions to help draw out student understanding
25-30 minutes
Compare the two diagrams below. How are they similar? How are they different?
Geometry - 8.8 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, 11:31 a.m..
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 $$F$$.
Geometry - 8.8 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, 11:33 a.m..
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.
A task that represents the peak thinking of the lesson - mastery will indicate whether or not objective was achieved
5-10 minutes
The radius of circle $$A$$ shown below has a value of 2.5 cm.
What is the length of $$\overline{AD}$$?
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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Construct tangent lines to a circle to define and describe the circumscribed angle.
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.
Standards
G.GPE.A.1G.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.1G.CO.A.5G.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.5G.GPE.A.1
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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.
Determine the angle and length relationships between intersecting chords.
Prove properties of angles in a quadrilateral inscribed in a circle.
G.C.A.3
G.C.A.2G.C.A.3
G.C.A.2G.C.A.4
Use angle and side length relationships with chords, tangents, inscribed angles, and circumscribed angles to solve problems.
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.
Calculate the sector area of a circle. Identify relationships between sector area, arc angle, and radius.
Use sector area of circles to calculate the composite area of figures.
G.C.B.5N.Q.A.2N.Q.A.3
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