Slide 1 / 255 Slide 2 / 255 Geometry Circles 2015-10-23 www.njctl.org Slide 3 / 255 Slide 4 / 255 Table of Contents Throughout this unit, the Standards for Mathematical Practice Click on a topic to go are used. to that section MP1: Making sense of problems & persevere in solving them. Parts of a Circle MP2: Reason abstractly & quantitatively. Central Angles & Arcs MP3: Construct viable arguments and critique the reasoning of others. Arc Length & Radians MP4: Model with mathematics. MP5: Use appropriate tools strategically. Chords, Inscribed Angles & Triangles MP6: Attend to precision. MP7: Look for & make use of structure. Tangents & Secants MP8: Look for & express regularity in repeated reasoning. Segments & Circles Additional questions are included on the slides using the "Math Practice" Pull-tabs (e.g. a blank one is shown to the right on Questions from Released PARCC Examination this slide) with a reference to the standards used. If questions already exist on a slide, then the specific MPs that the questions address are listed in the Pull-tab. Slide 4 (Answer) / 255 Slide 5 / 255 Throughout this unit, the Standards for Mathematical Practice are used. MP1: Making sense of problems & persevere in solving them. Parts of a Circle MP2: Reason abstractly & quantitatively. MP3: Construct viable arguments and critique the reasoning of Math Practice others. MP4: Model with mathematics. MP5: Use appropriate tools strategically. MP6: Attend to precision. MP7: Look for & make use of structure. MP8: Look for & express regularity in repeated reasoning. Additional questions are included on the slides using the "Math [This object is a pull tab] Practice" Pull-tabs (e.g. a blank one is shown to the right on this slide) with a reference to the standards used. Return to the If questions already exist on a slide, then the specific MPs that table of the questions address are listed in the Pull-tab. contents
Slide 6 / 255 Slide 7 / 255 Circles Circles are a type of Figure A figure lies in a plane and is contained by a boundary. Euclid defined a circle and its center in this way: Euclid defined figures in this way: Definition 15: A circle is a plane figure contained by one line such that all the straight lines falling upon it from one point among those lying within the figure equal one another. Definition 13: A boundary is that which is an extremity of anything. Definition 16: And the point is called the center of the circle. Definition 14: A figure is that which is contained by any boundary or boundaries. This states that all the radii (plural of radius) drawn from the A boundary divides a plane into those parts that are within the center of a circle are of equal length, which is a very boundary and those parts that are outside it. That which is within the important aspect of circles and their radii. boundary is the "figure." Slide 8 / 255 Slide 9 / 255 Circles and Their Parts Circles and Their Parts Another way of saying this is that a circle is made up of all The symbol for a circle is and is named by a capital letter the points that are an equal distance from the center of the placed by the center of the circle. circle. A The below circle is named: Circle A or radius A center AB is a radius of A radius (plural, radii ) is a line segment drawn from the center of the A circle to any point on the circle. circumference B Slide 10 / 255 Slide 11 / 255 Radii Diameters A radius (plural, radii ) is a line segment drawn from the center Definition 17: A diameter of the circle is any straight line drawn of the circle to any point on its circumference. through the center and terminated in both directions by the It follows from the definition of a circle that all the radii of a circumference of the circle, and such a straight line also bisects circle are congruent since they must all have equal length. the circle. An unlimited number of radii can be drawn in a circle. Since the diameter passes through the center of the circle and extends to the circumference on either side, it is twice the length of a radius of that circle. That all radii of a circle are congruent will be important to C solving problems. A In this drawing, we know that line segments AC, AD and AB B are all congruent. D
Slide 12 / 255 Slide 13 / 255 Diameters Chords A chord is a line segment whose endpoints lie on the There are an unlimited number of diameters which can be drawn circumference of the circle. within a circle. So, a diameter is a special case of a chord. They are all the same length, so they are all congruent. Why is a radius not a chord? F That all diameters of a circle E B D are congruent will be important F to solving problems. C All the line segments in this drawing A are chords. In this drawing, we know that G A line segments BE, CG and DF B are all congruent. G D E C Slide 13 (Answer) / 255 Slide 14 / 255 Chords Chords A chord is a line segment whose endpoints lie on the There are an unlimited number of chords which can be A radius starts at the center circumference of the circle. drawn in a circle. and has one endpoint on the circle. So, a diameter is a special case of a chord. Chords are not necessarily the same length, so are not necessarily congruent. Answer Why is a radius not a chord? A chord has 2 endpoints on the circle, w/ the center not being one of the endpoints. B B D D F F Question on this slide All the line segments in this drawing Chords can be of any length up to a addresses MP7. are chords. maximum. [This object is a pull tab] A A What is the longest chord that can be G G drawn in a circle? E E C C Slide 14 (Answer) / 255 Slide 15 / 255 Semicircles Chords There are an unlimited number of chords which can be Definition 18: A semicircle is the figure contained by the drawn in a circle. diameter and the circumference cut off by it. And the center of Chords are not necessarily the same length, so are not the semicircle is the same as that of the circle. necessarily congruent. BC, the diameter, is Answer the longest chord. The question on this diameter B D slide addresses MP7. semicircles F Chords can be of any length up to a maximum. [This object is a pull tab] A What is the longest chord that can be G drawn in a circle? E C
Slide 16 / 255 Slide 16 (Answer) / 255 Diameters and R adii Diameters and R adii AC = 5 The measure of the diameter, d , is twice the measure of The measure of the diameter, d , is twice the measure of DC = 10 the radius, r . the radius, r . Additional Q's to address MP Standards: In this case, CD = 2 AB In this case, CD = 2 AB Answer What information are you given? (MP1) What is the problem asking? (MP1) In general, d = 2r or r = 1/2 d In general, d = 2r or r = 1/2 d What do you think the answers will be? (MP2) Can you do this mentally? (MP1 & MP5) Example: In the diagram to the Example: In the diagram to the D D How can you check your answers? left, AB = 5. Determine AC & DC. left, AB = 5. Determine AC & DC. (MP1) [This object is a pull tab] A A B B C C Slide 17 / 255 Slide 17 (Answer) / 255 1 A diameter of a circle is the longest chord of the circle. 1 A diameter of a circle is the longest chord of the circle. True True False False Answer True [This object is a pull tab] Slide 18 / 255 Slide 18 (Answer) / 255 2 A radius of a circle is a chord of a circle. 2 A radius of a circle is a chord of a circle. True True False False Answer False [This object is a pull tab]
Slide 19 / 255 Slide 19 (Answer) / 255 3 The length of the diameter of a circle is equal to twice the 3 The length of the diameter of a circle is equal to twice the length of its radius. length of its radius. True True Answer False False True [This object is a pull tab] Slide 20 / 255 Slide 20 (Answer) / 255 4 If the radius of a circle measures 3.8 meters, what 4 If the radius of a circle measures 3.8 meters, what is is the measure of the diameter? the measure of the diameter? Answer 7.6 m [This object is a pull tab] Slide 21 / 255 Slide 21 (Answer) / 255 5 How many diameters can be drawn in a circle? 5 How many diameters can be drawn in a circle? A 1 A 1 B 2 B 2 C C Answer 4 4 D D D infinitely many infinitely many [This object is a pull tab]
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