Curriculum / Math / 10th Grade / Unit 7: Circles / Lesson 10
Math
Unit 7
10th Grade
Lesson 10 of 14
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Use angle and side length relationships with chords, tangents, inscribed angles, and circumscribed angles to solve problems.
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.
The essential concepts students need to demonstrate or understand to achieve the lesson objective
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This lesson summarizes key concepts in the unit; therefore, this lesson should be used as a review day.
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Problems designed to teach key points of the lesson and guiding questions to help draw out student understanding
In circle $$A$$ below, $$\overleftrightarrow{CE}$$ and $$\overleftrightarrow{DE}$$ are tangent at points $$C$$ and $$D$$, respectively.
Find $$m\angle{CED}$$ and $$m\angle{ADG}$$.
Geometry - 8.10 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, 11:55 a.m..
In the following diagram, the radius of circle $$D$$ is $$5$$ cm and $$F$$ is the midpoint of $$\overline{AE}$$. The measures of $$\widehat{EB}$$ and $$\widehat{BC}$$ are given in the diagram. Find the measures of all other unmarked angles, arcs, and segments.
Module 7: Circles a Geometric Perspective from Geometry: A Learning Cycle Approach made available by Mathematics Vision Project under the CC BY 4.0 license. © 2016 Mathematics Vision Project. Accessed Oct. 19, 2017, 2:58 p.m..
A task that represents the peak thinking of the lesson - mastery will indicate whether or not objective was achieved
In the diagram below, $$\overline{PA}$$ is tangent to circle $$O$$, and $$\overline{AB}$$ is a chord. If $$m\widehat{ACB}=300$$, find the measure of $$\angle BAP$$.
G.C.A.2: Chords, Secants, and Tangents 17 is made available on JMAP by Steve Sibol and Steve Watson. Copyright © 2017 JMAP, Inc. - All rights reserved. Accessed Sept. 19, 2018, 10:43 a.m..
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.
Lesson 9
Lesson 11
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
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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
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
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.5 N.Q.A.2 N.Q.A.3
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