1. Construct a large circle and label its centre C. D Construct a chord AB and a central angle BCA. Measure BCA.

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1 10.1 ploring ngles in a ircle Focus on fter this lesson, ou will be able to describe a relationship between inscribed angles in a circle relate the inscribed angle and central angle subtended b the same arc Madd, Jennifer, and Nia are each about to shoot at the empt net. If the are each equall accurate with their shot, who do ou think is most likel to score? Materials compass or circular geoboard with elastic bands protractor ruler chord a line segment with both endpoints on a circle plore Relationships etween ngles in a ircle 1. onstruct a large circle and label its centre. onstruct a chord and a central angle. Measure. 2. reate the inscribed angle. What is the measure of? 3. How do the measures of and compare? central angle an angle formed b two radii of a circle inscribed angle an angle formed b two chords that share a common endpoint arc (of a circle) a portion of the circumference chord central angle arc 4. reate a second inscribed angle. What is the measure of? 5. hoose another point on the circle between and. reate one more inscribed angle that has its arms touching the endpoints of the arc. What is the measure of this inscribed angle? 6. Repeat steps 1 to 5 for a different sized circle, and a different sized chord. Reflect and heck 7. a) What is the relationship between a central angle and an inscribed angle that stands on the same arc? b) What is the relationship between all the inscribed angles that stand on the same arc? inscribed angle 8. Predict which hocke plaer in the opening paragraph is most likel to score on the empt net. plain. 378 MHR hapter 10

2 Link the Ideas You can use properties related to angles in a circle to solve problems. Inscribed ngles The inscribed angles subtended b the same arc are congruent. entral and Inscribed ngles The measure of the central angle is equal to twice the measure of the inscribed angle subtended b the same arc. ample 1: etermine ngle Measures in a ircle Point is the centre of the circle. = 35 a) What is the measure of? Justif our answer. b) What is the measure of? Justif our answer. Solution a) The inscribed angles, and, are equal because the are subtended b the same arc,. Therefore, = 35. b) The central angle is subtended b the same arc as the inscribed angle. central angle is twice the measure of an inscribed angle that is subtended b the same arc. = 2 = 2 35 = 70 Therefore, = Literac Link n angle that subtends an arc or a chord is an angle that "stands on" or is formed b the endpoints of the arc or chord. Web Link You ma wish to eplore these geometric properties on a geoboard or on a computer. o to www. mathlinks9.ca and follow the links. Show You Know Point is the centre of the circle. = 55. What are the measures of angles and? Justif our answers ploring ngles in a ircle MHR 379

3 ample 2: Use entral and Inscribed ngles to Recognize Relationships Point is the centre of the circle. diameter = 10 cm chord = 6 cm a) What is the measure of? plain our reasoning. b) What is the length of the chord? Justif our answer. 6 cm 10 cm Solution a) The diameter divides the circle into two semicircles. Since is a straight line, the central angle is 180. Then, must be half of 180 because it is an inscribed angle that is subtended b the same arc,. The measure of is 90. b) Since = 90, is a right triangle. The Pthagorean relationship can be used to find the length of = = = = 64 = 64 = 8 Therefore, = 8 cm. Show You Know Point is the centre of the circle. is the diameter. chord = 12 cm chord = 5 cm a) What is the measure of? plain our reasoning. b) What is the length of the diameter? 12 cm 5 cm 380 MHR hapter 10

4 ample 3: Use entral and Inscribed ngles to Solve Problems Jamie works for a realtor. One of his jobs is to photograph houses that are for sale. He photographed a house two months ago using a camera lens that has a 70 field of view. He has returned to the house to update the photo, but he has forgotten his original lens. Toda he onl has a telephoto lens with a 35 field of view. X 70 house From what location(s) could Jamie photograph the house, with the telephoto lens, so that the entire house still fills the width of the picture. plain our choices. Solution raw a circle with the centre located at the verte of the 70 angle. Use one arm of the angle as the radius of the circle. onstruct an number of different inscribed angles that each contains the front of the house. n of these points are locations from which Jamie could take the photo. The measure of each of these inscribed angles will be half the measure of the central angle = 35 ach inscribed angle will measure 35, which corresponds to the field of view for Jamie s telephoto lens. epending on access, and whether there are an trees or a garden in the wa, an point on the major arc that is outside of the house will work. 35 X 70 house Strategies raw a iagram Identif all Possibilities Literac Link major arc is more than a semicircle. minor arc is less than a semicircle. Show You Know flashlight has a field of view measuring 25, and a camera has a field of view measuring 50. How can ou position the camera and flashlight so that the camera will capture the same area as the flashlight illuminates? 10.1 ploring ngles in a ircle MHR 381

5 Ke Ideas Inscribed angles subtended b the same arc of a circle are equal. = central angle is twice the measure of an inscribed angle subtended b the same arc. = 2 n inscribed angle is one half the measure of a central angle subtended b the same arc. = 1 2 When the inscribed angle is subtended b a diameter of the circle, the inscribed angle is equal to 90. Literac Link The identical markings at the inscribed angles and indicate that the measures of these angles are equal. heck Your Understanding ommunicate the Ideas 1. In the diagram, measures half of. oes the rule for inscribed angles hold true for?plain our reasoning.? 2. Mann constructed a circle using a compass. He used a straight edge to draw a diameter. Then, he constructed an inscribed angle that shared endpoints with the diameter. What is the measure of the inscribed angle he constructed? How do ou know? Practise For help with #3 to #5, refer to ample 1 on page What are the measures of and? Justif our answers a) What is the measure of FJ? plain our reasoning. b) What is the measure of F? Justif our answer. F H J MHR hapter 10

6 5. raw a circle with a central angle that measures 60. raw and label the measure of two inscribed angles that are subtended b the same arc as the central angle. For help with #6 and #7, refer to ample 2 on page Point is the centre of the circle. diameter = 17 cm chord = 15 cm 9. For a high school drama production, three spotlights are positioned on an arc at the back of the theatre, just above the audience. ach spotlight projects light through an angle of 22 and fills the rectangular front of the stage. Use a diagram to identif an ideal location to take a photo of the performance using a camera with a lens that has a field of view of cm 15 cm a) What is the measure of? plain. b) What is the length of the chord? 7. The circle has centre and a radius of 8 cm. F = 45. F 8 cm 45 a) What is the measure of F? b) What is the length of the chord F? press our answer to the nearest tenth of a centimetre. ppl 10. In the diagram, is the centre of the circle and = 38. For each of the following questions, justif our answer. a) What is the measure of? b) What tpe of triangle is? c) What is the measure of? 38 For help with #8 and #9, refer to ample 3 on page fter a power outage, Jacob helps his mother b shining a flashlight beam at the breaker panel while she locates the tripped breakers. His flashlight projects light through an angle of 15, while his mother s flashlight projects light through an angle of 30. Use a diagram to show a good place for Jacob to stand so that his flashlight will illuminate the same area of the breaker panel as his mother s flashlight does. 11. Point is the centre of the circle and F = 25. Justif each of our answers to the following questions. a) What is the measure of F? b) What is the measure of F? 25 F 10.1 ploring ngles in a ircle MHR 383

7 12. If KJM = 15, JML = 24, and point is at the centre of the circle, what is the measure of each of the following angles? J 15 L 24 a) KLM b) JKL c) JL d) KM 13. In the diagram, = 34 and = a) What is the measure of? b) What is the measure of? c) What tpe of triangle is? d) What is the measure of? 14. fter looking at the diagram of the circle, manda decides to use the Pthagorean relationship to calculate the length of chord. Will this method work? plain. 55 K M 15. Find the unknown angle measures, and, in each diagram. Where is labelled, it is the centre of the circle. a) b) c) d) esign a geometr question involving a given central angle for which the answer is an inscribed angle measuring 30. Include a diagram with our question. 17. circle with centre has a diameter. The inscribed angle measures 14. What are the measures of and? raw a diagram. tend 18. Find the unknown angle measures, and, in each diagram, given that is the centre of the circle. a) b) m 4 m 384 MHR hapter 10

8 19. hole has a diameter of 20 cm. What is the maimum side length of a square that will fit into the hole? 20 cm 21. In the semicircle, H = 27. is on the diameter and is the midpoint of For each of the following diagrams, calculate the value of. a) b) F H etermine the measure of each angle, justifing our work mathematicall. a) H b) H c) d) e) a) esign a piece of art using one circle and an number of inscribed and central angles. b) escribe how the angles and line segments in our design are related. Tech Link Inscribed and entral ngles In this activit, ou will use dnamic geometr software to eplore inscribed and central angles in a circle. To use this activit, go to and follow the links. plore 1. a) What is the measure of the central angle? b) What is the measure of the inscribed angle? c) What is the measure of the minor arc? 2. rag point around the circle. What happens to the measure of the two angles O and? Wh does this happen? 3. rag either point or point around the circle. Record at least four measurements of the inscribed angle and the O central angle from different locations on the circle. 4. escribe an relationships between the central angle O and the inscribed angle subtended b the same arc ploring ngles in a ircle MHR 385