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DG4GP_905_06.qxd 12/27/06 10:23 AM Page 25 CHAPTER 6 Discovering and Proving Circle Properties Content Summary In Chapter 6, students continue to build their understanding of geometry as they explore properties of circles. Some of these properties are associated with line segments related to circles; other properties are associated with arcs and angles. A circle is defined as a set of points equidistant from a fixed point, its center. Line Segments Related to Circles The best-known line segments related to a circle are its radius and diameter. Actually, the word radius can refer either to a line segment between a point on the circle and the center, or to the length of such a line segment. Similarly, diameter means either a line segment that has endpoints on the circle and passes through the center, or the length of such a line segment. The diameter is a special case because it is the longest chord of a circle; a chord is a line segment whose endpoints are on the circle. Another line segment associated with circles is a tangent segment, which touches the circle at just one point and lies on a tangent line, which also touches the circle at just one point and is perpendicular to the radius at this point. Students learned about these segments in Chapter 1, and Lesson 6.1 provides a quick review. Tangent Radius Diameter Chord Arcs and Angles A piece of the circle itself is an arc. If you join each endpoint of an arc to the center of the circle, you form the central angle that intercepts the arc. The size of the arc can be expressed in degrees—the number of degrees in the arc’s central angle. This chapter explores several such relationships among arcs, angles, and segments. Students also write paragraph and flowchart proofs to confirm the universality of these relationships. The size of the arc can also be expressed in length. The arc length is calculated by using the total circumference, or the distance around the circle. The number is defined to be the circumference of any circle divided by that circle’s diameter; or, the circumference is times the diameter. For example, if an arc is 14 of the complete circle, then its central angle measures 14 of 360°, and its length is 14 of the circle’s circumference. Summary Problem Draw a diagram of a central angle intercepting a chord of a circle and its arc, as shown in the picture. A Move points A, B, and C to different locations to illustrate the ideas of the chapter. Questions you might ask in your role as student to your student: ● ● ● C What concepts are illustrated in the original drawing? How could you move each of the points A, B, and C to show an inscribed angle? How could you move each of the points A, B, and C to show tangent segments? B (continued) ©2008 Key Curriculum Press Discovering Geometry: A Guide for Parents 25 DG4GP_905_06.qxd 12/27/06 10:23 AM Page 26 Chapter 6 • Discovering and Proving Circle Properties (continued) ● ● How could you move points A, B, and C to show an angle inscribed in a semicircle? How could you move points A, B, and C to show parallel lines intercepting congruent arcs? Sample Answers The original drawing shows a central angle, a sector, a chord, and an arc. If the center, C, is moved to lie on the circle, an inscribed angle is formed. and BC If C is moved to be outside the circle, and A and B are moved to make AC tangents, then those segments are congruent. Or, the chord in the original diagram could be rotated at one of its ends until it becomes a tangent segment. is a diameter and B remains on the circle, ABC is a right If C is moved until AC angle inscribed in a circle. You would need to add another point and put C on the circle to show two chords. Parallel chords intercept equal arcs if they are equidistant from the center. C A B Of course many other answers are possible. Encourage your student to think of multiple ways that A, B, and C could be moved to illustrate these same concepts. 26 Discovering Geometry: A Guide for Parents ©2008 Key Curriculum Press DG4GP_905_06.qxd 12/27/06 10:23 AM Page 27 Chapter 6 • Review Exercises Name Period Date , find mOAB, mAOB, 1. (Lesson 6.1) Given tangent AB and mABO. A B C O 310 2. (Lessons 6.2, 6.3) Find the unknown measures or lengths. b 10 cm a c d 130 . 3. (Lesson 6.3) ABC is an equilateral triangle. Find mAB A B C 4. (Lesson 6.4) Write a paragraph proof to prove the following: B and BD . Given: Circle A with diameters EC BC Prove: ED E A C D 5. (Lessons 6.5, 6.7) Given that the circumference of circle A is . 24 in., find the radius of the circle and the length of BDC C B 120⬚ A D and BC are tangents to the circle as shown. 6. (Lessons 6.1, 6.3) AB ED . Find a and b. AC B 15 m 100⬚ b C A D a E 140⬚ ©2008 Key Curriculum Press Discovering Geometry: A Guide for Parents 27 DG4GP_905_06.qxd 12/27/06 10:23 AM Page 28 SOLUTIONS TO CHAPTER 6 REVIEW EXERCISES 1. mAOB 90° 360° 310° 50° mAC 50° mAOC mAC 50° 90° mABO 180° mABO 40° 2. Central angle is equal to arc measure. Triangle sum. Subtraction. Diameter is perpendicular to the chord. d 10 cm Diameter perpendicular to a chord bisects the chord. AB BC AC mAC mAB mBC mBC mAC 360° mAB mAB mAB 360° mAB 360° 3mAB 120° mAB 28 360° in a circle. b 90° c 180° 130° 50° 1 a 2(130°) ma 65° 3. Tangent is perpendicular to radius. 180° in a semicircle. Inscribed angles. and BD on circle A intersect to form 4. Diameters CE congruent vertical angles, so mBAC mDAE. The measure of an arc equals the measure of its mED because the central angle. Therefore, mBC measures of their central angles are equal. 5. C 2r 24 in.; therefore, r 12 in. 360° 120° 240° mBDC 240 length of BDC 360 (24) 16 in. 6. mCD mAE a Parallel secants intercept congruent arcs. 100° a a 140° 360° 360° in a circle. a 60° Solve. b 15 m Tangent segments from the same point are congruent. Division. Equilateral triangle. Congruent arcs of congruent chords. 360° in a circle. Substitution. Combine like terms. Division. Discovering Geometry: A Guide for Parents ©2008 Key Curriculum Press