Circles

9 Components
Initial Task
Square Peg
Big Ideas
Big Idea 1 Big Idea 2 Big Idea 3
Formative Assessment Lesson
Inscribing and Circumscribing Right Triangles
Re-engagement
Re-engagement for Unit 7
End of Unit Assessments
End of Unit Assessment for Unit 7 End of Unit Assessment Teacher Resources for GEO Unit 7 Find aligned Regents questions with Quiz Banker
Geo U6: Circles
Browse Components
Initial Task
Square Peg
Big Ideas
Big Idea 1 Big Idea 2 Big Idea 3
Formative Assessment Lesson
Inscribing and Circumscribing Right Triangles
Re-engagement
Re-engagement for Unit 7
End of Unit Assessments
End of Unit Assessment for Unit 7 End of Unit Assessment Teacher Resources for GEO Unit 7 Find aligned Regents questions with Quiz Banker
Big Idea:

Big Idea 3

Congruence and similarity criteria prove relationships between segments and figures of a circle.

1 week
See Standards
Common Core: Major Standards:
G-MG.A.1

Apply geometric concepts in modeling situations. Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder).

G-SRT.B.5

Prove theorems involving similarity. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Clarification: ASA, SAS, SSS, AAS, and Hypotenuse-Leg theorem are valid criteria for triangle congruence. AA, SAS, and SSS are valid criteria for triangle similarity.

G-SRT.C.8

Define trigonometric ratios and solve problems involving right triangles. Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.★

G-MG.A.3

Apply geometric concepts in modeling situations. Apply geometric methods to solve design problems (e.g., designing an object or structure to satisfy physical constraints or minimize cost; working with typographic grid systems based on ratios).★

Common Core: Supporting Standards:
G-CO.D.12

Make geometric constructions. Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). Copying a segment; copying an angle; bisecting a segment; bisecting an angle; constructing perpendicular lines, including the perpendicular bisector of a line segment; and constructing a line parallel to a given line through a point not on the line. Clarification: Constructions include but are not limited to the listed constructions. Example: constructing the median of a triangle or constructing an isosceles triangle with given lengths.

G-CO.D.13

Make geometric constructions. Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle.

Common Core: Additional Standards:
G-C.A.3

Understand and apply theorems about circles. Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle.

G-C.A.2

Understand and apply theorems about circles. 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.
Clarification: Relationships include but are not limited to the listed relationships. Example: angles involving tangents and secants.

Evidence of Understanding

  • investigate and describe the relationships that exist between the segments on a circle
    • investigate and describe how the lengths of segments for two intersecting chords, two secants (drawn from the same point), or a tangent and a secant (drawn from the same point) are related
      • prove theorems about the segments of secants, tangents, or intersecting chords
    • recognize congruent chords are equidistant from the center of the circle
    • recognize tangents from a shared point outside the circle are congruent
    • calculate the length of an unknown segment
  • analyze and apply characteristics of polygons that circumscribe or are inscribed in a circle
    • use constructions to visualize the congruent relationships when inscribing or circumscribing a polygon in the circle
      • construct an equilateral triangle, square or regular hexagon inscribed in a circle
      • construct a right triangle inscribed in a circle using perpendicular bisectors
      • construct a circle that is inscribed in a triangle using angle bisectors
    • use constructions to identify and define the incenter of a triangle
    • prove properties of angles for a quadrilateral inscribed in a circle
      • Example: quadrilaterals inscribed in a circle have opposite angles that are supplementary
    • find the missing value of an unknown angle or segment involving a polygon and a circle
  • use circles in modeling situations and find missing values
    • create diagrams that can be used to represent the situation
    • use markings and notations to create equations that can be used to find missing values

Develop conceptual understanding:

chord, secant, tangent, secants, inscribed, circumscribed, incenter 

Supporting terms to communicate:

radius, equivalent, ratio, proportion, similarity, scale factor, dilation, right triangle, equilateral, regular polygon, quadrilateral, perpendicular,  bisect, intersect, intercepted arc, supplementary

Search for resources related to this big idea by following this link.

Core Resource
A core resource supports multiple days of instruction.
  • Circles and Segments
    These four parts are designed to support students in making connections between chord, secant, and tangent line relationships in circles and creating proofs of these relationships using triangle similarity.
    Resource:
    Circles and Segments

    Geometry & Algebra II Archive

    Geo U6: Circles

    Big Idea: Big Idea 3

    These four parts are designed to support students in making connections between chord, secant, and tangent line relationships in circles and creating proofs of these relationships using triangle similarity.

    Preview ResourceAdd a Copy of Resource to my Google Drive
    Type
    Geometry: Circles
    File
    Google Doc
    Tags
    activity, chord, inscribed, tangent line
    Standards
    Common Core: Major Standards ( G-SRT.B.5, G-SRT.C.8, G-MG.A.1, G-MG.A.3 ) Commo…
    +— Details
    Common Core: Major Standards:
    G-SRT.B.5

    Prove theorems involving similarity. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Clarification: ASA, SAS, SSS, AAS, and Hypotenuse-Leg theorem are valid criteria for triangle congruence. AA, SAS, and SSS are valid criteria for triangle similarity.

    G-SRT.C.8

    Define trigonometric ratios and solve problems involving right triangles. Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.★

    G-MG.A.1

    Apply geometric concepts in modeling situations. Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder).

    G-MG.A.3

    Apply geometric concepts in modeling situations. Apply geometric methods to solve design problems (e.g., designing an object or structure to satisfy physical constraints or minimize cost; working with typographic grid systems based on ratios).★

    Common Core: Supporting Standards:
    G-CO.D.12

    Make geometric constructions. Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). Copying a segment; copying an angle; bisecting a segment; bisecting an angle; constructing perpendicular lines, including the perpendicular bisector of a line segment; and constructing a line parallel to a given line through a point not on the line. Clarification: Constructions include but are not limited to the listed constructions. Example: constructing the median of a triangle or constructing an isosceles triangle with given lengths.

    G-CO.D.13

    Make geometric constructions. Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle.

    Common Core: Additional Standards:
    G-C.A.2

    Understand and apply theorems about circles. 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.
    Clarification: Relationships include but are not limited to the listed relationships. Example: angles involving tangents and secants.

    G-C.A.3

    Understand and apply theorems about circles. Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle.

    All Resources From:
Instructional Routine: Connecting Representations
These tasks are embedded within the instructional routine called Connecting Representations.