Curriculum / Math / 10th Grade / Unit 7: Circles / Lesson 7
Math
Unit 7
10th Grade
Lesson 7 of 14
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Prove properties of angles in a quadrilateral inscribed in a circle.
The core standards covered in this lesson
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.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
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Problems designed to teach key points of the lesson and guiding questions to help draw out student understanding
25-30 minutes
Below is a quadrilateral inscribed in a circle, also known as a cyclic quadrilateral.
Describe the relationship between the opposite angles $$\angle CDB$$ and $$\angle CEB$$ in the cyclic quadrilateral using the intercepted arcs.
Geo_U7_L7_AP1 by Kerry Taylor is made available by GeoGebra under the CC BY-NC-SA 3.0 license. Copyright © International GeoGebra Institute, 2013. Accessed Sept. 19, 2018, 1:45 p.m..
Below is the same cyclic quadrilateral as shown in Anchor Problem #1. Show, using the radii marked and the angle measures marked, the relationship between opposite angles.
Geometry - 8.7 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, 11:22 a.m..
A task that represents the peak thinking of the lesson - mastery will indicate whether or not objective was achieved
5-10 minutes
What value of $$x$$ guarantees that the quadrilateral shown in the diagram below is cyclic? Explain your reasoning.
Geometry > Module 5 > Topic E > Lesson 20 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..
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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Define and determine properties of tangents and secants of circles to solve problems with inscribed and circumscribed triangles.
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.
G.C.A.3
G.C.A.2G.C.A.3
Construct tangent lines to a circle to define and describe the circumscribed angle.
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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