Curriculum / Math / 10th Grade / Unit 7: Circles / Lesson 1
Math
Unit 7
10th Grade
Lesson 1 of 14
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Derive the equation of a circle using the Pythagorean Theorem where the center of the circle is at the origin.
The core standards covered in this lesson
G.GPE.A.1 — Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle given by an equation.
G.GPE.B.4 — Use coordinates to prove simple geometric theorems algebraically. For example, prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, √3) lies on the circle centered at the origin and containing the point (0, 2).
The foundational standards covered in this lesson
G.GPE.B.7 — Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula. Modeling is best interpreted not as a collection of isolated topics but in relation to other standards. Making mathematical models is a Standard for Mathematical Practice, and specific modeling standards appear throughout the high school standards indicated by a star symbol (★). The star symbol sometimes appears on the heading for a group of standards; in that case, it should be understood to apply to all standards in that group.
8.G.B.7 — Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-world and mathematical problems in two and three dimensions.
8.G.B.8 — Apply the Pythagorean Theorem to find the distance between two points in a coordinate system.
The essential concepts students need to demonstrate or understand to achieve the lesson objective
Suggestions for teachers to help them teach this lesson
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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
Write an equation that models each point’s distance from the center of the circle, point $$A$$.
Geometry-8.1 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 7, 2017, 9:05 a.m..
A circle has a center at the origin and contains the point ($$2, \space \sqrt{5}$$).
A task that represents the peak thinking of the lesson - mastery will indicate whether or not objective was achieved
5-10 minutes
The circle shown below has a center at the origin.
If a point ($$3, \space y$$) lies on the circle, what is the value of $$y$$?
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.
Next
Given a circle with a center translated from the origin, write the equation of the circle and describe its features.
Topic A: Equations of Circles
Standards
G.GPE.A.1G.GPE.B.4
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
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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