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G.C.1-4 2011 Domain Circles Cluster Understand and apply theorems about circles Standards 1. Prove that all circles are similar. 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. 3. Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle. 4. (+) Construct a tangent line from a point outside a given circle to the circle. Essential Questions Enduring Understandings Why are all circles similar? How can you prove relationships between angles in a circle? When lines intersect a circle, or within a circle, how do you find the measures of resulting angles? What is the relationship between inscribed and circumscribed figures? Students will understand the different components of circles and how they relate to angle measurements. Students will understand that inscribed angles on a diameter are right angles. Students will understand that the radius of a circle is perpendicular to the tangent where the radius intersects the circle. Activities, Investigation, and Student Experiences 1. Interactive examples, PowerPoint lessons, regular level and challenge level problems, sketchpad activities, and hands on activities 2. Video and Interactive Examples 3. Sample Lessons and Examples 4. SMARTboard Slides with examples G.C.1-4 Content Statements Students will be able to use the fact that the ratio of diameter to circumference is the same for circles to prove that all circles are similar. Students will be able to use definitions, properties, and theorems, to identify and describe relations among inscribed angles, radii, and chords (including central, inscribed, and circumscribed angles.) Students will be able to construct inscribed circles of a triangle. Students will be able to construct circumscribed circles of a triangle. Students will be able to use definitions, properties, and theorems to prove properties of angles for a quadrilateral inscribed in a circle. Students will be able to 2011 G.C.1-4 find the measure of an inscribed angle. Students will be able to find the measure of an angle formed by a tangent and a chord. Students will be able to construct a tangent line from a point outside a given circle to the circle. Assessments Student Participation Questioning Quizzes (Teacher Given and Self Quizzes): o http://teachers.henrico.k12.va.us/math/igo/ 08Circles/8-1Terminology/8-1t.htm o http://teachers.henrico.k12.va.us/math/igo/ 08Circles/8-3Tangents/8-3t.htm o http://teachers.henrico.k12.va.us/math/igo/ 08Circles/8-4ArcsChords/8-4t.htm o http://teachers.henrico.k12.va.us/math/igo/ 08Circles/8-5AngleFormulas/8-5t.htm o http://teachers.henrico.k12.va.us/math/igo/ 08Circles/8-6SegmentFormulas/8-6t.htm Benchmark/Test (Click here for resource folder) http://sites.google.com/site/ppshighschoolmath/ho me/benchmark-assessments/assessment Homework 2011 G.C.1-4 Equipment Needed: SMARTboard Projector Paper and pencil Calculator Compass Ruler Protractor Geometer’s Sketchpad 2011 Teacher Resources: 1. Henrico County Public Schools: http://teachers.henrico.k12.va.us/math/igo/08Circles/8_1.htm l http://teachers.henrico.k12.va.us/math/igo/08Circles/8_3.htm l http://teachers.henrico.k12.va.us/math/igo/08Circles/8_4.htm l http://teachers.henrico.k12.va.us/math/igo/08Circles/8_5.htm l http://teachers.henrico.k12.va.us/math/igo/08Circles/8_6.htm l 2. Math Warehouse: http://www.mathwarehouse.com/geometry/circle/interactivecentral-angle-of-circle.php http://www.mathwarehouse.com/geometry/circle/inscribedangle.html#inscribedAngleDemo http://www.mathwarehouse.com/geometry/circle/tangent-tocircle.php 3. Click here for Resource Folder. 4. Click here for Resource Folder.