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Segments of Chords, Secants, and Tangents CK12 Editor Say Thanks to the Authors Click http://www.ck12.org/saythanks (No sign in required) To access a customizable version of this book, as well as other interactive content, visit www.ck12.org CK-12 Foundation is a non-profit organization with a mission to reduce the cost of textbook materials for the K-12 market both in the U.S. and worldwide. Using an open-content, web-based collaborative model termed the FlexBook®, CK-12 intends to pioneer the generation and distribution of high-quality educational content that will serve both as core text as well as provide an adaptive environment for learning, powered through the FlexBook Platform®. Copyright © 2012 CK-12 Foundation, www.ck12.org The names “CK-12” and “CK12” and associated logos and the terms “FlexBook®” and “FlexBook Platform®” (collectively “CK-12 Marks”) are trademarks and service marks of CK-12 Foundation and are protected by federal, state, and international laws. Any form of reproduction of this book in any format or medium, in whole or in sections must include the referral attribution link http://www.ck12.org/saythanks (placed in a visible location) in addition to the following terms. Except as otherwise noted, all CK-12 Content (including CK-12 Curriculum Material) is made available to Users in accordance with the Creative Commons Attribution/NonCommercial/Share Alike 3.0 Unported (CC BY-NC-SA) License (http://creativecommons.org/licenses/by-nc-sa/3.0/), as amended and updated by Creative Commons from time to time (the “CC License”), which is incorporated herein by this reference. Complete terms can be found at http://www.ck12.org/terms. Printed: February 19, 2013 AUTHOR CK12 Editor www.ck12.org C ONCEPT Concept 1. Segments of Chords, Secants, and Tangents 1 Segments of Chords, Secants, and Tangents Learning Objectives • Find the lengths of segments associated with circles. In this section we will discuss segments associated with circles and the angles formed by these segments. The figures below give the names of segments associated with circles. Segments of Chords Theorem If two chords intersect inside the circle so that one is divided into segments of length a and b and the other into segments of length b and c then the segments of the chords satisfy the following relationship: ab = cd. This means that the product of the segments of one chord equals the product of segments of the second chord. Proof We connect points A and C and points D and B to make 4AEC and 4DEB. 1 www.ck12.org 6 6 6 AEC ∼ = 6 DEB ∼ 6 BDC CAB = ACD ∼ = 6 ABD Vertical angles Inscribed angles intercepting the same arc Inscribed angles intercepting the same arc Therefore, 4AEC ∼ 4DEB by the AA similarity postulate. In similar triangles the ratios of corresponding sides are equal. c a = ⇒ ab = cd b d Example 1 Find the value of the variable. 10x = 8 × 12 10x = 96 x = 9.6 Segments of Secants Theorem If two secants are drawn from a common point outside a circle and the segments are labeled as below, then the segments of the secants satisfy the following relationship: a(a + b) = c(c + d) 2 www.ck12.org Concept 1. Segments of Chords, Secants, and Tangents This means that the product of the outside segment of one secant and its whole length equals the product of the outside segment of the other secant and its whole length. Proof We connect points A and D and points B and C to make 4BCN and 4ADN. 6 6 BNC ∼ = 6 DNA NBC ∼ = 6 NDA Same angle Inscribed angles intercepting the same arc Therefore, 4BCN ∼ 4DAN by the AA similarity postulate. In similar triangles the ratios of corresponding sides are equal. a c+d = ⇒ a(a + b) = c(c + d) c a+b Example 2 Find the value of the variable. 3 www.ck12.org 10(10 + x) = 9(9 + 20) 100 + 10x = 261 10x = 161 x = 16.1 Segments of Secants and Tangents Theorem If a tangent and a secant are drawn from a point outside the circle then the segments of the secant and the tangent satisfy the following relationship a(a + b) = c2 . This means that the product of the outside segment of the secant and its whole length equals the square of the tangent segment. Proof We connect points C and A and points B and C to make 4BCD and 4CAD. 4 www.ck12.org Concept 1. Segments of Chords, Secants, and Tangents m6 CDB = m6 BAC − m6 DBC The measure of an Angle outside a circle is equal to half the difference of the measures of the intercepted arcs m6 BAC = m6 ACD + m6 The measure of an exterior angle in a triangle equals CDB the sum of the measures of the remote interior angles m6 CDB = m6 ACD + m6 CDB − m6 DBC Combining the two steps above m6 DBC = m6 ACD algebra Therefore, 4BCD ∼ 4CAD by the AA similarity postulate. In similar triangles the ratios of corresponding sides are equal. a c = ⇒ a(a + b) = c2 a+b c Example 3 Find the value of the variable x assuming that it represents the length of a tangent segment. x2 = 3(9 + 3) x2 = 36 x=6 5 www.ck12.org Lesson Summary In this section, we learned how to find the lengths of different segments associated with circles: chords, secants, and tangents. We looked at cases in which the segments intersect inside the circle, outside the circle, or where one is tangent to the circle. There are different equations to find the segment lengths, relating to different situations. Review Questions 1. Find the value of missing variables in the following figures: a. b. 6 www.ck12.org Concept 1. Segments of Chords, Secants, and Tangents c. d. e. 7 www.ck12.org f. g. h. 8 www.ck12.org Concept 1. Segments of Chords, Secants, and Tangents i. j. 9 www.ck12.org k. l. 10 www.ck12.org Concept 1. Segments of Chords, Secants, and Tangents m. n. o. 11 www.ck12.org p. q. r. 12 www.ck12.org Concept 1. Segments of Chords, Secants, and Tangents s. t. 2. A circle goes through the points A, B,C, and D consecutively. The chords AC and BD intersect at P. Given that AP = 12, BP = 16, and CP = 6, find DP? 3. Suzie found a piece of a broken plate. She places a ruler across two points on the rim, and the length of the chord is found to be 6 inches. The distance from the midpoint of this chord to the nearest point on the rim is found to be 1 inch. Find the diameter of the plate. 4. Chords AB and CD intersect at P. Given AP = 12, BP = 8, and CP = 7, find DP. Review Answers 1. a. b. c. d. e. f. g. h. i. j. 14.4 16 4.5 32.2 12 29.67 4.4 18.03 4.54 20.25 13 www.ck12.org k. 7.48 l. 23.8 m. 24.4 n. 9.24 or 4.33 o. 17.14 p. 26.15 q. 7.04 r. 9.8 s. 4.4 t. 8 2. 4.5 3. 10 inches. 4. 13.71 Texas Instruments Resources In the CK-12 Texas Instruments Geometry FlexBook, there are graphing calculator activities designed to supplement the objectives for some of the lessons in this chapter. See http://www.ck12.org/flexr/chapter/9694. 14