Geometry (& GT)
Unit 3: Circles, Proofs, and Constructions
In this unit, students will build on their understanding of similarity to investigate relationships between circles. In addition, students will explore and prove relationships between parts of circles, to include radii, tangents, secants, and chords. Students should understand how these parts relate to segment lengths and angle measures, and how this relates back to similarity. Through multiple constructions, students will explore properties of other figures and how this relates to circles. Students will justify the formulas for circumference and area, and use them to explore arc length, define radians, and derive the formula for the area of a sector. Using their understanding of the Cartesian coordinate system, students will use distance formula to write equations of circles given a radius and center. Students should be able to justify whether or not a given point lies on a given circle using their understanding of coordinate geometry.
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What will my child learn?
Students will:
Understand and apply theorems about circles.
G.C.A.1
Prove that all circles are similar.
Understand and apply theorems about circles.
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.
Understand and apply theorems about circles. (Geometry GT)
G.C.A.4 (+)
Construct a tangent line from a point outside a given circle to the circle.
Understand and apply theorems about circles.
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.
Make geometric constructions.
G.CO.D.13
Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle.
Explain volume formulas and use them to solve problems.
G.GMD.A.1
Give an informal argument for the formulas for the circumference of a circle, area of a circle. Use dissection arguments and informal limit arguments.
Find arc lengths of sectors of circles.
G.C.B.5
Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector (SAT® Content  ATM.05  ATM.06).
Translate between the geometric description and the equation for a conic section.
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 (SAT® Content  ATM.07).
Use coordinates to prove simple geometric theorems algebraically.
G.GPE.B.4
Use coordinates to prove simple geometric theorems algebraically, i.e. prove or disprove that the point lies on the circle centered at the origin and containing the point (0, 2).
Students will:
Understand and apply theorems about circles.
G.C.A.1
Prove that all circles are similar.
 Background Info.
 Check for Understanding: Defining Similarity through AnglePreserving Transformations
 Review/Rewind: Proof: All Circles are Similar
 Enrichment Tasks: Similar Circles
Understand and apply theorems about circles.
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.
 Background Info.
 Check for Understanding: Central, Inscribed, and Circumscribed Angles  Inscribed Angles 1
 Review/Rewind: Inscribed Angles
 Enrichment Tasks: Right Triangles Inscribed in Circles I  Right Triangles Inscribed in Circles II
Understand and apply theorems about circles. (Geometry GT)
G.C.A.4 (+)
Construct a tangent line from a point outside a given circle to the circle.
 Check for Understanding: Constructing a Line Tangent to a Circle
 Review/Rewind: Arc Length from Subtended Angle: Radians
 Enrichment Tasks: Tangent to a Circle From a Point
Understand and apply theorems about circles.
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.
 Background Info.
 Check for Understanding: Central, Inscribed, and Circumscribed Angles  Inscribed Shapes
 Review/Rewind: Inscribe Quadrilaterals Proof
 Enrichment Tasks: Opposite Angles in a Cyclic Quadrilateral  Circumscribed Triangles
Make geometric constructions.
G.CO.D.13
Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle.
 Check for Understanding: Inscribed Shapes in a Circle
 Review/Rewind: Proof: Right Triangles Inscribed in Circles
 Enrichment Tasks: Inscribing an Equilateral Triangle in a Circle  Inscribing a Square in a Circle
Explain volume formulas and use them to solve problems.
G.GMD.A.1
Give an informal argument for the formulas for the circumference of a circle, area of a circle. Use dissection arguments and informal limit arguments.
 Background Info.
 Check for Understanding: Solid Real Life Word Problems
 Review/Rewind: Volume of a Cylinder  Volume of a Cone
 Enrichment Tasks: Volume of a Special Pyramid  Volume Formulas for Cylinders and Prisms
Find arc lengths of sectors of circles.
G.C.B.5
Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector (SAT® Content  ATM.05  ATM.06).
 Check for Understanding: Areas of Circles and Sectors  Circles and Arcs
 Review/Rewind: Arc Length as Fraction of Circumference
 Enrichment Tasks: Mutually Tangent Circles
Translate between the geometric description and the equation for a conic section.
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 (SAT® Content  ATM.07).
 Background Info.
 Check for Understanding: Pythagorean Theorem and Radii of a Circle  Equation of a Circle in Factored Form
 Review/Rewind: Writing Standard Equation of a Circle
 Enrichment Tasks: Slopes and Circles  Explaining the Equation For a Circle
Use coordinates to prove simple geometric theorems algebraically.
G.GPE.B.4
Use coordinates to prove simple geometric theorems algebraically, i.e. prove or disprove that the point lies on the circle centered at the origin and containing the point (0, 2).
 Check for Understanding: Geometry Problems on the Coordinate Plane
 Review/Rewind: Challenge Problem: Points of Two Circles
 Enrichment Tasks: Midpoint Miracle
What are some signs of student mastery?

Tools & Technology
Math Open Reference  a free interactive tool that allows students to explore Geometry through constructions and animations. GeoGebra  a dynamic math software tool that allows users to explore geometry, algebra, tables, graphing, statistics and other areas of math in one easytouse package. It can be downloaded for FREE on a computer or tablet device. 
More 4 U
These videos show how to use tools to:
Looking for clarification on some of the vocabulary used in the Geometry course? A slight variation in units, click here to download the MD State Department of Education's (MSDE's) geometry glossary. 