Curriculum / Math / 10th Grade / Unit 7: Circles / Lesson 4
Math
Unit 7
10th Grade
Lesson 4 of 14
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Define a chord to derive the Chord Central Angles Conjecture and Thales’ Theorem.
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 foundational standards covered in this lesson
7.G.B.4 — Know the formulas for the area and circumference of a circle and use them to solve problems; give an informal derivation of the relationship between the circumference and area of a circle.
7.G.B.5 — Use facts about supplementary, complementary, vertical, and adjacent angles in a multi-step problem to write and solve simple equations for an unknown angle in a figure.
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
Based on the definition of each feature of a circle below, label the circle with the appropriate vocabulary words.
Arc: Part of any curve Central angle: An angle formed by two radii with the vertex as the center of the circle Chord: A line segment containing two points on a curve Diameter: A line segment connecting two points on a curve that passes through the center of the circle Radius: A line segment that connects the center of a circle and a point on the circle Tangent: A line that intersect a circle at one and only one point
Geometry - 8.4 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, 11:02 a.m..
Below is circle $$A$$ with points $$E$$, $$C$$, $$D$$, and $$B$$ on the circle. $$\angle EAC \cong \angle DAB$$.
What is the relationship between $$\overline{BD}$$ and $$\overline {EC}$$?
Geometry - 8.4 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, 11:04 a.m..
Given diameter $$\overline{CB}$$, chords $$\overline{DB}$$ and $$\overline{CD}$$, and radii $$\overline{AC}$$, $$\overline{AD}$$, and $$\overline{AB}$$, find the measure of $$\angle CDB$$.
Geometry - 8.4 AP3 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, 11:07 a.m..
A task that represents the peak thinking of the lesson - mastery will indicate whether or not objective was achieved
5-10 minutes
Circle $$A$$ is shown below.
Geometry > Module 5 > Topic A > Lesson 1 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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Describe the relationship between inscribed and central angles in terms of their intercepted arc.
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
G.C.A.2
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.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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