Circles Angles and Arcs

Transcription

1 Math Objectives Students will know the definitions of and identify central angles, major and minor arcs, intercepted arcs, and inscribed angles of a circle. Students will determine and apply the following relationships: Two inscribed angles intercepting the same arc have the same measure. An inscribed angle measure of 90 results in the endpoints of the intercepted arc lying on a diameter. The measure of an angle inscribed in a circle is half the measure of the central angle that intercepts the same arc. Students will construct viable arguments and critique the reasoning of others (CCSS Mathematical Practice). Students will look for and express regularity in repeated reasoning (CCSS Mathematical Practice). Vocabulary central angle inscribed angle major, minor, and intercepted arc About the Lesson This lesson involves manipulating endpoints of an arc, manipulating an inscribed angle, and manipulating the vertex of an angle intercepting an arc. As a result students will: Use visualization to understand the definitions of central angle, intercepted arc, and minor and major arcs. Infer that the sum of the measures of minor and major arcs is 360, that two inscribed angles intercepting the same arc have the same measure, and that the inscribed angle has half the measure of the central angle that intercepts the same arc. Deduce that the opposite angles of a quadrilateral inscribed in a circle are supplementary. TI-Nspire Navigator System Use Screen Capture to examine examples of systems of inequalities. Use Teacher Edition computer software or Live Presenter to review student documents and discuss examples as a class. TI-Nspire Technology Skills: Download a TI-Nspire document Open a document Move between pages Grab and drag a point Attach segments to an angle and an angle to a segment Tech Tips: Make sure the font size on your TI-Nspire handheld is set to Medium. Press / G to either hide the function line or access the function line in a Graphs & Geometry page. Lesson Materials: Student Activity Circles_Angles_and_Arcs_ Student.pdf Circles_Angles_and_Arcs_ Student.doc TI-Nspire document Circles_Angles_and_Arcs.tns Visit for lesson updates and tech tip videos Texas Instruments Incorporated 1 education.ti.com

2 Use Quick Poll to assess student understanding throughout the lesson. Discussion Points and Possible Answers Tech Tip: If students experience difficulty dragging a point, check to make sure that they have moved the cursor until it becomes a hand ( ) getting ready to grab the point. Press / x to grab the point and close the hand ({). Move to page Drag point A or point C. Describe the changes that occur in the figure as you drag the point. Answer: The angle measurement changes but is never more than 180. The solid part of the circle is always in the interior of the angle. Tech Tip: The dashed attribute will not show on the handheld; that part of the circle will show up as gray instead of black. 2. Angle AOC is called a central angle. Why do you think this is so? Answer: The vertex is at the center of the circle. An angle intercepts an arc of a circle if each endpoint of the arc is on a different ray of the angle and the other points of the arc are in the interior of the angle. Move to page 1.3. As you move point A or point C, the central angle AOC intercepts a minor arc AC. The measure of the minor arc equals the measure of the central angle. The larger remaining arc, ABC, is called a major arc Texas Instruments Incorporated 2 education.ti.com

3 Teacher Tip: On the TI-Nspire, arcs are measured using Length, not Angle. Therefore, if a student uses the built-in measuring tool, TI-Nspire will report arc length rather than arc measure. 3. a. Move point A or point C to help you complete the table. Sample answer: The completed table is below. The final row of students tables will vary. AOC arc AC arc ABC arc AB + arc ABC (Choose an angle.) b. What is true about the measure of AB + ABC, the sum of the measures of the minor and major arcs? Answer: The measures of AB + ABC always equals In a circle, the measure of a central angle AOC is n. a. What is the measure of the minor arc that is intercepted by the central angle? How do you know? Answer: The minor arc measures n because an intercepted arc has the same measurement as its central angle. b. What is the measure of the major arc? How do you know? Answer: The major arc measures (360 n) because the sum of the measures of the major and minor arcs is 360. TI-Nspire Navigator Opportunity: Quick Poll See Note 1 at the end of this lesson Texas Instruments Incorporated 3 education.ti.com

4 Move to page Angle ABC is called an inscribed angle because BA and BC are chords of the circle and vertex B is on the circle. Drag point B around the circle. a. As point B is moved around the circle, what do you notice about the measure of ABC? Answer: Angle ABC has the same measure until it intercepts the other arc. While point B is moved around on that arc, ABC will remain the same. Teacher Tip: Some students may recognize that the two angles measures sum to 180. b. Why does m ABC change when point B is moved from one arc to the other? Explain your reasoning. Answer: The angle measure changes because the intercepted arc changes. The intercepted arc is always in the interior of the inscribed angle. c. Move point A or point C until ABC is a right angle. What is special about the arc and AC? Answer: The arc measure is 180 and the arc is a semicircle. AC is a diameter. Teacher Tip: When students reach the right angle, the diameter should show up as a dotted segment. TI-Nspire Navigator Opportunity: Screen Capture See Note 2 at the end of this lesson. Move to page 1.5. Angle ABC intercepts arc AC. Drag point D to various locations outside the circle, on the circle, inside the circle, and at the center O. 6. Place point D on the circle so that ADC intercepts the same arc as ABC. a. What do you notice about the measures of ABC and ADC? 2011 Texas Instruments Incorporated 4 education.ti.com

5 Answer: The angle measures are the same. The angles are congruent. The ratio of the angle measurements is 1. TI-Nspire Navigator Opportunity: Quick Poll See Note 3 at the end of this lesson. Teacher Tip: Make sure both angles intercept the same arc. Some students may incorrectly place point D on the (bold) arc intercepted by ABC. Point D should snap to the circle when it gets close. b. What happens to the angles if you move point A or point C? Answer: The angle measures change. The angles are congruent and the ratio of the measure of ADC to the measure of ABC is 1 if the angles intercept the same arc. The angles are not congruent and the ratio is not 1 if the angles do not intercept the same arc. TI-Nspire Navigator Opportunity: Screen Capture See Note 4 at the end of this lesson. Teacher Tip: Some students may note that the angle measurements are the same whenever AD and BC intersect or criss-cross and are not the same when they don t. This observation is important but needs to be related to intercepted arcs. Some students may notice that the angles are supplementary when they do not intercept the same arc. This property is addressed in problem 9. If m ABC equals 90, then m ADC equals 90 whether or not they share the same intercepted arc. 7. Place point D at the center of the circle. Move point A and point C so that ADC intercepts the same arc as ABC. a. What is the relationship between the measures of inscribed ABC and central ADC? Answer: The measure of the central angle is double the measure of the inscribed angle. The measure of the inscribed angle is half the measure of the central angle. The ratio of the measure of ADC to the measure of ABC is 2. TI-Nspire Navigator Opportunity: Quick Poll See Note 5 at the end of this lesson. b. What happens to the angles if you move point A or point C? Answer: The measure of the inscribed angle is half the measure of the central angle (the ratio of the measure of ADC to the measure of ABC is 2), as long as both angles intercept the same 2011 Texas Instruments Incorporated 5 education.ti.com

6 arc. TI-Nspire Navigator Opportunity: Screen Capture and/or Live Presenter See Note 6 at the end of this lesson. Teacher Tip: For a proof of the Inscribed Angle Theorem: In the simplest case, one leg of the inscribed angle is a diameter of the circle so it passes through the center of the circle. Since that leg is a straight line, the supplement of the central angle equals 180 2θ. Drawing a segment from the center of the circle to the other point of intersection of the inscribed angle produces an isosceles triangle, made from two radii of the circle and the second leg of the inscribed angle. Since two angles in an isosceles triangle are equal and since the angles in a triangle sum to 180, it follows that the inscribed angle equals θ, half of the central angle. 8. Leona said, Since a central angle can never measure more than 180, I know an inscribed angle can never measure more than 90. Do you agree or disagree? Why? Answer: I disagree because the central angle always intercepts a minor arc, but an inscribed angle can intercept a major arc. 9. Place point D on the circle so that ABCD is a quadrilateral. a. What do you notice about the sum of the measures of ABC and ADC? Check with a classmate to compare. Answer: The sum of the measures of ABC and ADC is 180. b. What do you notice about the sum of the measures of the angles if you move point A or point C? Answer: As long as ABCD remains a quadrilateral, the sum of the measures of ABC and ADC remains 180. c. What do you notice about arcs ABC and ADC? Answer: One is a major arc and one is a minor arc. Together the arcs make a circle. The measures of the arcs sum to 360. d. How does the relationship between arcs ABC and ADC explain the sum of the measures of inscribed ABC and ADC? Answer: The sum of the measures of the major and minor arcs is 360. Since the measures of the inscribed angles are half the measures of their intercepted arcs, the angles are supplementary Texas Instruments Incorporated 6 education.ti.com

7 Teacher Tip: A quadrilateral inscribed in a circle has the special name cyclic quadrilateral. TI-Nspire Navigator Opportunity: Quick Poll See Note 7 at the end of this lesson. Wrap Up: Upon completion of the discussion, the teacher should ensure that students understand: Two inscribed angles intercepting the same arc have the same measure. An inscribed angle measure of 90 results in the endpoints of the intercepted arc lying on a diameter. The measure of the inscribed angle is half the measure of the central angle that intercepts the same arc. TI-Nspire Navigator Note 1 Questions 4a and 4b, Quick Poll: Have students enter their answers to the first part of questions 4a and 4b in Quick Poll. Ask students to answer, How Do You Know? orally. Note 2 Question 5c, Screen Capture: As students complete question 5c, use Screen Capture to view their screens. Discuss the following: Are all the right angles located in exactly the same place? What is true about the arc intercepted by the right angle? What is the measure of the intercepted arc? What is AC on each screen if a right angle is shown? Note 3 Question 6a, Quick Poll: Using the Open Response feature of Quick Poll, have students enter their answer to 6a. Note 4 Question 6b, Screen Capture: Use Screen Capture to look at students screens when they complete question 6b. Put the screens together where the angles intercept the same arc and the screens together where the angles do not intercept the same arc. Discuss the results Texas Instruments Incorporated 7 education.ti.com

8 Note 5 Question 7a, Quick Poll: Using Quick Poll, students answer the following: The ratio of the measure of ADC to the measure of ABC is. Discuss students responses. Note 6 Question 7b, Screen Capture: Use Screen Capture to discuss students answers to question 7b. Screen Capture can also be used to help students answer the extension questions. Note 7 Wrap Up, Quick Poll: Use Quick Poll to be sure students understand the three statements listed under Wrap Up. The True/False feature would be a good choice Texas Instruments Incorporated 8 education.ti.com

9 Application of a Circle Angles and Arcs Math Objectives Students will identify and know the difference between central angles and inscribed angles of a circle. Students will identify the relationships between the measures of inscribed angles and the arcs they intercept. Students will identify and know the difference between major arcs and minor arcs. Students will identify and find the diameter, circumference, and area of a circle given the appropriate formulas. Students will see an example of how circles can be used in a realworld application. Vocabulary central angle inscribed angle major arc minor arc diameter radius circumference area About the Lesson This lesson is a follow-up lesson to the activity Circles Angles and Arcs. This lesson involves using relationships among different types of angles and arcs in a circle to solve a real-world application involving the design of a courtyard. Students will use basic circle concepts, such as inscribed angles, to manipulate the design to desired specifications. Students will use other circle concepts, such as diameter, radius, circumference, and area, to determine basic information needed to construct the courtyard. TI-Nspire Navigator TM System Use Teacher Edition computer software to review student documents. TI-Nspire Technology Skills: Download a TI-Nspire document Open a document Move between pages Grab and drag a point Find the length of a segment Find the length of a circle Find the area of a circle Create a segment Tech Tips: Make sure the font size on your TI-Nspire handhelds is set to Medium. Lesson Materials: Student Activity Application_of_a_Circle _Angles_and_Arcs _Student.pdf Application_of_a_Circle _Angles_and_Arcs _Student.doc TI-Nspire document Application_of_a_Circle _Angles_and_Arcs.tns Visit for lesson updates and tech tip videos Texas Instruments Incorporated 1 education.ti.com

10 Application of a Circle Angles and Arcs Discussion Points and Possible Answers Tech Tip: If students experience difficulty dragging a point, check to make sure that they have moved the cursor until it becomes a hand ( ) getting ready to grab the point. Also, be sure that the word point appears, not the word text. Then press / x to grab the point and close the hand ({). Move to page 1.2. Suppose you work for an architectural firm and a new business complex is in the process of being designed. The plans for the complex include a circular courtyard within a square area with side lengths of 8.4 yards. The courtyard will use 10-inch square pavers in different colors to create a design, as shown in the diagram on page 1.4. The points A, B, C, D, and E represent the points of the star design, and each of the points lies on the circle. Point Q represents the center of the circle. You have been asked to supply the company constructing the courtyard some information that will help with creating the design and ordering supplies. Move to page Would the angles A, B, C, D, and E be considered central angles or inscribed angles? Explain. Answer: The angles would be inscribed angles since the vertices are on the circle and the sides of the angles are chords of circle Q. The vertex of a central angle is at the center of the circle. 2. What is the relationship between the measures of the angles A, B, C, D, and E and the arcs they intercept? Explain. Answer: A central angle has the same degree measure as the arc it intercepts. An inscribed angle is half the degree measure of the central angle. Therefore, the degree measures of the inscribed angles of circle Q are half the measures of the arcs they intercept Texas Instruments Incorporated 2 education.ti.com

11 Application of a Circle Angles and Arcs Teacher Tip: Students may need help coming to this conclusion. Providing a hint such as using the relationship between the measures of central angles and the arcs they intercept may be helpful. 3. Would the arcs intercepted by each of the angles A, B, C, D, and E be considered major arcs or minor arcs? Explain. Answer: A major arc is a part of a circle that measures between 180 and 360 degrees. A minor arc measures less than 180 degrees. Therefore, the arcs intercepted by each of the angles would be minor arcs. 4. In order for the star pattern to be uniform, each of the angles should have the same degree measure. What should be the degree measure of each of the angles A, B, C, D, and E? Explain your reasoning. Answer: Since 360 degrees represents the circle and there are 5 angles, divide 360 by 5 to get 72 degrees. An inscribed angle is half the measure of the intercepted arc. Divide 72 by 2 to get 36 degrees. Each of the angles should measure 36 degrees. 5. Grab and move points A, B, C, D, and E around the circle until all angle measures are the same. What is the degree measure? Is this what you expected? Explain. Sample Answer: The degree measure of each of the five angles is 36 degrees. This result will correspond with the answer to Question 4, if students answered correctly. 6. What is the diameter of circle Q? Explain your reasoning. Answer: By looking at the diagram and visualizing a diameter of circle Q that is parallel with two sides of the square, it appears as though the diameter will be the same as the length of the sides of the square in which it is inscribed. Check your answer to Question 6 by using the Segment tool (MENU > Points & Lines > Segment) to create a segment that represents the diameter of circle Q. Use the Length tool (MENU > Measurement > Length) to find the length of the segment. Change the Attributes (MENU > Actions > Attributes) of the measurement to one decimal place Texas Instruments Incorporated 3 education.ti.com

12 Application of a Circle Angles and Arcs 7. Given that the circumference of a circle can be found by using the formula C = 2πr, where r is the radius, find the circumference of circle Q to the nearest yard. Show your work below. Answer: C = 2πr = 2 π 4.2 = = 26 yd Check your answer to Question 7 by using the Length tool to find the length of circle Q. Change the Attributes of the measurement to display zero decimals. 8. Given that the area of a circle can be found by using the formula A = πr 2, where r is the radius, find the area of circle Q to the nearest yard. Show your work below. Answer: A = πr 2 = π 4.22 = = 55 yd 2 Check your answer to Question 8 by using the Area tool (MENU > Measurement > Area) to find the area of circle Q. Change the Attributes of the measurement to display zero decimals. 9. Given that the pavers being used to construct the courtyard are squares with side lengths of 10 inches, how many pavers will need to be ordered to construct the courtyard? (1 yd 2 = 1,296 in. 2 ) Answer: First, convert the area of the circle to inches, since the pavers are given in units of inches, and then divide by 100 since the pavers are 10-inch squares. At least 713 pavers would need to be ordered to construct the courtyard. 2 1,296 in yd = 71,280 in in. 2 = yd 2010 Texas Instruments Incorporated 4 education.ti.com

13 Application of a Circle Angles and Arcs Wrap Up Upon completion of the discussion, the teacher should ensure that students understand: The difference between central angles and inscribed angles of a circle. The relationship between the measure of inscribed angles and the arcs they intercept. The difference between a major arc and a minor arc. How to identify the diameter of a circle. The relationship between the diameter and radius of a circle. How to find the circumference and area of a circle. Assessment Complete a similar problem by changing the dimensions of the square in which the circle is inscribed, or have students design their own courtyard using both inscribed and central angles Texas Instruments Incorporated 5 education.ti.com