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Name_____________________________________ Class____________________________ Date ________________ Lesson 12-2 Lesson Objectives 1 Use congruent chords, arcs, and central angles 2 Recognize properties of lines through the center of a circle Chords and Arcs NAEP 2005 Strand: Geometry Topic: Relationships Among Geometric Figures Local Standards: ____________________________________ All rights reserved. Vocabulary and Key Concepts. Theorem 12-4 Within a circle or in congruent circles (1) Congruent central angles have (2) Congruent chords have (3) Congruent arcs have chords. arcs. central angles. Theorem 12-5 Within a circle or in congruent circles (1) Chords equidistant from the center are © Pearson Education, Inc., publishing as Pearson Prentice Hall. (2) Congruent chords are . from the center. Theorem 12-6 In a circle, a diameter that is perpendicular to a chord bisects the and its . Theorem 12-7 In a circle, a diameter that bisects a chord (that is not a diameter) is to the chord. Theorem 12-8 In a circle, the perpendicular bisector of a chord contains the of the circle. A chord is P Q O Daily Notetaking Guide Geometry Lesson 12-2 231 Name_____________________________________ Class____________________________ Date________________ Examples. A 1 Using Theorem 12-4 In the diagram, radius OX bisects AOB. What can you conclude? AOX by the definition of an angle bisector. AX because central angles have X O chords. because chords have arcs. B 2 Using Theorem 12-5 Find AB. QS QR RS Segment Addition Postulate QS Substitute. QS Simplify. AB Chords that are equidistant from the center of a circle are congruent. AB Substitute 7 The distance from the center of O to PQ is measured along a perpendicular line. PM ( 2 ) P r O Use the Pythagorean Theorem. 2 Substitute. Simplify. r 232 M Substitute. r2 The radius of O is S A diameter that is perpendicular to a chord bisects the chord. OP 2 PM 2 OM 2 r2 7 Q The distance from O to PQ is 15 in., and PQ 16 in. Find the radius of O. PQ R for QS. 3 Using Diameters and Chords P and Q are points on O. PM 4 C 4 A Q Find the square root of each side. in. Geometry Lesson 12-2 All rights reserved. B Daily Notetaking Guide © Pearson Education, Inc., publishing as Pearson Prentice Hall. 0 AX Name_____________________________________ Class____________________________ Date ________________ Quick Check. 1. If you are given that BC DF in the circles, what can you conclude? B O D P C F All rights reserved. O P 2. Find the value of x in the circle at the right. 18 18 16 x 36 © Pearson Education, Inc., publishing as Pearson Prentice Hall. 3. Use the circle at the right. a. Find the length of the chord to the nearest unit. 6.8 4 x b. Find the distance from the midpoint of the chord to the midpoint of its minor arc. Daily Notetaking Guide Geometry Lesson 12-2 233