<meta http-equiv="refresh" content="1; url=/nojavascript/"> Circles | CK-12 Foundation

# 3.9: Circles

Created by: CK-12

I. Section Objectives

• Distinguish between radius, diameter, chord, tangent, and secant of a circle.
• Find relationships between congruent and similar circles.
• Examine inscribed and circumscribed polygons.
• Write the equation of a circle.

II. Cross- curricular-Nature

• Look at the following examples of circles in nature.
• These images are Figure 09.01.01
• www.naturesmightypictures.blogspot.com/2006/06/circles-in-nature.html
• While these don’t specifically name all of the parts of a circle, use these images to discover the different parts of a circle.
• Where is the radius or the diameter?
• Is there a polygon inscribed in any of the circles?
• For example, look at the sunflower or the rose.
• Are any of the circles similar?
• For example, look at the patterns in the different images. Do you see any similar circles?
• Have a discussion with the students that broadens their thinking about circles and the parts of a circle.
• Then ask the students to find an example of circles in nature.
• Bring it into class the next day.

III. Technology Integration

• Students are going to be working to make connections between circles and real world activities.
• How are circles used in different careers?
• This first example is a designer who makes wheels.
• This designer makes wheels that are used in performance racing.
• As students watch this video, have them make notes on the different geometric elements that are mentioned in the video.
• Then following the video, conduct a discussion on how geometry and wheel design are related.
• www.thefutureschannel.com/hands-on_math/spoke_math.php

IV. Notes on Assessment

• Assessment is completed through class discussion.
• Work to have all students participate in the discussion.
• Ask questions of the students and provide feedback as needed.

## Tangent Lines

I. Section Objectives

• Find the relationship between a radius and a tangent to a circle.
• Find the relationship between two tangents draw from the same point.
• Circumscribe a circle.
• Find equations of concentric circles.

II. Cross- curricular-Design

• Have students look at the following image from Wikipedia.
• This is Figure 09.02.01
• www.en.wikipedia.org/wiki/File:Concentric_(PSF).png
• This is a picture of concentric circles.
• Have students discuss the characteristics of concentric circles.
• Are they similar?
• How can we design a concentric circle?
• Ask the students to create a black and white art design using concentric circles.
• Students will need white paper, black pencil or marker, a compass.
• Have the students work to create their own design.
• Also have them insert one tangent line somewhere in the design.
• Allow time for students to share their work when finished.

III. Technology Integration

• This is a very fascinating website for students to explore.
• www.cut-the-knot.org/Curriculum/Geometry/TangentTwoCirclesI.shtml
• In working with this website, students will be manipulating the center of one of the circles.
• They can click on the center and drag the center anywhere that they wish to.
• When they do this, they will alter the diagram of the two circles and their tangents.
• It is a great visual and very interactive.

IV. Notes on Assessment

• Assess student work with the art design.
• Is the design of the concentric circles accurate?
• Are the circles organized around a common center?
• Is there a tangent in the design?
• Does the student understand what a tangent is based on what he/she has drawn?
• Provide students with feedback as needed.

## Common Tangents and Tangent Circles

I. Section Objectives

• Solve problems involving common internal tangents of circles.
• Solve problems involving common external tangents of circles.
• Solve problems involving externally tangent circles.
• Solve problems involving internally tangent circles.
• Common tangent

• Use the following images from Wikipedia on the Mad Tea Party.
• This is Figure 09.03.01
• Students can even complete the technology integration first to see some real pictures of the tea cups in action.
• Tell the students that their task is to draw the design of the Mad Tea Party using circles that are connected.
• The design of the Mad Tea Party consists of three small turntables, which rotate counter clock-wise, each holding six teacups, within one large turntable, rotating clockwise.
• Students are to draw this design and how they hypothesize that the circles are or are not connected.
• Do the students think that tangents play a role in this?
• Why or why not?
• Ask the students to write a short paragraph explaining their thinking about the ride.

III. Technology Integration

• Have students complete a websearch for the Mad Tea Party at Disney World.
• Students will see images and can even see a film clip of the ride on youtube.
• Students can use this information to assist them in drawing the design of the ride.

IV. Notes on Assessment

• Assess student work.
• How did the students draw the design of the ride?
• What was the student’s hypothesis about tangents?
• Does the reasoning make sense?
• Provide students with comments on their work.

## Arc Measures

I. Section Objectives

• Measure central angles and arcs of circles.
• Find relationships between adjacent arcs.
• Find relationships between arcs and chords.

II. Cross-curricular-Plate Design

• Use the image of the dinner plate with the stripes by Cynthia Rowley.
• This is Figure 09.04.01
• www.prontohome.com/product/whim-by-cynthia-rowley-melamine-p_1213285046
• Use this to show the students where there are arcs and chords.
• Then show them major and minor arcs as well.
• The assignment is for the students to design their own plate design using lines, chords and circles.
• Tell the students that they are free to design the plate however they would like as long as they label the major arcs and the minor arcs.
• They also need to figure out the measure of one of the arcs and explain how they completed this task.
• Allow time for the students to share their work when finished.

III. Technology Integration

• Have students go to www.brittanica.com the encyclopedia Brittanica’s website and search for Eratosthenes of Cyrene.
• Have them research how he used arcs to figure out the circumference of the earth.
• This may be challenging for the students to understand, so you may want to either allow them to work in small groups or to discuss this as a whole class.
• Begin by having them take notes on their own, then begin the discussion.

IV. Notes on Assessment

• Assess each plate design.
• Is it creative?
• Does it use the concepts of chords and arcs?
• Are the major and minor arcs labeled?
• Did the students figure out the measure of one of the arcs?
• Is the work written out and explained?
• Provide students with comments/feedback on their work.

## Chords

I. Section Objectives

• Find the lengths of chords in a circle.
• Find the measure of arcs in a circle.

II. Cross- curricular-Archimedes

• Use the image and information at the following website.
• Display this image for the students to see.
• This is Figure 09.05.01
• Now have the students copy this image on a piece of paper.
• In small groups, the students need to use this image to prove that the sum of the intercepted opposite arcs is equal.
• Students can use the text to refer back to the information that they have learned.
• They need to write five statements that demonstrate that this is a true statement.
• Students should be prepared to present their findings.

III. Technology Integration

• Begin with this statement, “It could be said that a spoke is the chord of a wheel.”
• Use different wheel designs to demonstrate how this is true or untrue.
• You may use a drawing program to draw and design support for your answer.
• You may also use a collection of images to support your answer.
• Be prepared to share your work when finished.

IV. Notes on Assessment

• When the students present their findings, listen to their reasoning.
• Challenge the others in the class to do the same thing.
• How does it support what we know about perpendicular lines and angles?
• How does it support what we have learned about arcs?
• Does the reasoning of the group make sense or is something missing?
• Is there a diagram to support their thinking?
• Did the students complete any measurements?
• Provide students with feedback on their work.

## Inscribed Angles

I. Section Objectives

• Find the measure of inscribed angles and the arcs they intercept.

II. Cross- curricular-Theater

• This is a problem that needs to be solved. It will require the students to use angle measures.
• This is picture of a seating chart for the Fichander Theater.
• This is Figure 09.05.01
• www.gotickets.com/venues/dc/fichandler_theatre.php
• Be sure that each student has a copy of the image.
• Show students how this is a theater in the round.
• The seating is arranged in a circle.
• The students need to use what they have learned about angles and arcs to determine which seats have the best angle to see the stage.
• Note: Students may determine right away that all of the seats are equal due to their angles. Why is this? Have them prove their thinking.

III. Technology Integration

• Students can go to the following website for a worksheet where they can practice finding the measure of inscribed angles.
• This is a great site for simple practice and drill of skills already learned.
• www.regentsprep.org/Regents/math/geometry/GP15/PcirclesN.htm
• Students can also go to any of several websites to find further explanation of inscribed angles and of the measure of those angles.
• Any of these sites will support students in expanding their understanding.

IV. Notes on Assessment

• Walk around and observe students as they work.
• Then have the students share their thinking about the theater problem.
• Be sure students are able to articulate their reasoning by using content from geometry.
• Diagrams are an excellent way for students to share their thinking.

## Angles of Chords, Secants and Tangents

I. Section Objectives

• Find the measures of angles formed by chords, secants and tangents.

II. Cross- curricular-Poetry

• Students are assigned the task of writing a poem or rap about the theorems in the text.
• Students can choose to write their poem about one of the theorems or all of the theorems.
• Students could also write a poem that defines and explains the relationship between chords, secants and tangents.
• It isn’t necessary to give too many directions for this assignment.
• Let the students work in small groups, and they will illustrate their level of understanding of the material through the poem.
• When finished, allow students time to present their work.

III. Technology Integration

• Have students complete this chapter by completing a websearch on circles in architecture.
• They can google this topic.
• Have the students keep track of the sites that they visit.
• They need to select three different images that best illustrates the content of the chapter.
• The students need to write a paragraph explaining how each one illustrates the concepts of the chapter, and which concepts it illustrates.
• Have students share their work when finished.

IV. Notes on Assessment

• Assess student work through their presentations.
• How well does the poem explain the theorem or theorems?
• How well does the poem explain the definitions from the text?
• Are the images that the student selected in line with the content from the chapter?
• Did the student explain which concepts are illustrated in the image?
• Is the information accurate?
• Provide students with feedback on their work.

## Segments of Chords, Secants and Tangents

I. Section Objectives

• Find the lengths of segments associated with circles.

II. Cross- curricular- Circus math

• This is a problem having to do with the circus.
• Here is the problem.
• A circus ring has a diameter of $42\;\mathrm{feet}$.
• A high wire is stretched across the diameter of the circle
• A second wire is stretched across the diameter of the circle.
• The two wires intersect at one point.
• On the first wire, the lengths of the wire are ten feet and eight feet.
• On the second wire, only one section of the wire is known and that is five feet.
• What is the length of the second section of the wire?
• Have students work in small groups on this problem.
• It is a great idea to have students draw a diagram of the solution of the problem.
• Solution:
• $10 \times 8 = 5x$
• $80 = 5x$
• $x = 16\;\mathrm{feet}$
• The diameter of the circle has no impact on the answer of this problem.

III. Technology Integration

• Have students go to the following website to do some research about high wire acts in the circus.
• www.reachoutmichigan.org/funexperiments/agesubject/lessons/newton/hwire.html
• What kind of math is involved in this art?
• Does the diameter of the wire impact the act?
• Have students write a short report on what they have learned about math and the high wire.
• Students can even complete the activity at the end of the web page and experience walking a “high wire” of sorts themselves.

IV. Notes on Assessment

• Check the solution to the problem.
• Did the students use a diagram?
• Is the diagram accurate?
• Were they able to solve the problem accurately?
• Provide students with feedback and comments.

## Date Created:

Feb 22, 2012

Feb 23, 2012
You can only attach files to None which belong to you
If you would like to associate files with this None, please make a copy first.