10.6 Find Segment Lengths
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1 10. Find Segment Lengths in ircles Goal p Find segment lengths in circles. Your Notes VOULRY Segments of a chord Secant segment Eternal segment THEOREM 10.14: SEGMENTS OF HORS THEOREM If two chords intersect in the interior of a circle, then the product of the lengths of the segments of one chord is equal to the product of the E lengths of the segments of the other chord. E p 5 E p Eample 1 Find lengths using Theorem Find ML and JK. NK p NJ 5 p p ( 1 5) 5 ( ) p ( ) Find ML and JK by substitution. L 1 3 N K 1 5 M J 1 1 ML 5 ( ) 1 ( ) JK 5 1 ( ) opyright Holt Mcougal. ll rights reserved. Lesson 10. Geometry Notetaking Guide 285
2 10. Find Segment Lengths in ircles Goal p Find segment lengths in circles. Your Notes VOULRY Segments of a chord When two chords intersect in the interior of a circle, each chord is divided into two segments called segments of the chord. Secant segment secant segment is a segment that contains a chord of a circle, and has eactly one endpoint outside the circle. Eternal segment n eternal segment is the part of a secant segment that is outside the circle. THEOREM 10.14: SEGMENTS OF HORS THEOREM If two chords intersect in the interior of a circle, then the product of the lengths of the segments of one chord is equal to the product of the E lengths of the segments of the other chord. E p E 5 E p E Eample 1 Find lengths using Theorem Find ML and JK. NK p NJ 5 NL p NM p ( 1 5) 5 ( 1 3 ) p ( 1 1 ) Find ML and JK by substitution. L 1 3 N K 1 5 M J 1 1 ML 5 ( 1 1 ) 1 ( 1 3 ) JK 5 1 ( 1 5 ) opyright Holt Mcougal. ll rights reserved. Lesson 10. Geometry Notetaking Guide 285
3 THEOREM 10.15: SEGMENTS OF SENTS THEOREM If two secant segments share the same endpoint outside a circle, then the product of the lengths of one secant segment and its eternal segment equals the product of the lengths of the E other secant segment and its eternal segment. E p 5 E p Eample 2 Use Theorem Find the value of. T P 4 Q R 5 S RQ p RP 5 RS p RT Use Theorem. p ( 1 ) 5 p ( 1 ) Substitute. 5 1 Simplify. 5 Solve for. 1. heckpoint Find the value of Lesson 10. Geometry Notetaking Guide opyright Holt Mcougal. ll rights reserved.
4 THEOREM 10.15: SEGMENTS OF SENTS THEOREM If two secant segments share the same endpoint outside a circle, then the product of the lengths of one secant segment and its eternal segment equals the product of the lengths of the E other secant segment and its eternal segment. E p E 5 E p E Eample 2 Use Theorem Find the value of. T P 4 Q R 5 S RQ p RP 5 RS p RT Use Theorem p ( 1 4 ) 5 5 p ( 1 5 ) Substitute Simplify. 7 5 Solve for. 1. heckpoint Find the value of Lesson 10. Geometry Notetaking Guide opyright Holt Mcougal. ll rights reserved.
5 THEOREM 10.1: SEGMENTS OF SENTS N TNGENTS THEOREM If a secant segment and a tangent segment share an endpoint outside a circle, then the product of the E lengths of the secant segment and its eternal segment equals the square of the length of the tangent segment. E 2 5 p Eample 3 Find lengths using Theorem 10.1 Use the figure at the right to find RS. 14 R S 12 Q RQ 2 5 RS p RT Use Theorem. 2 5 p ( 1 ) Substitute Simplify Write in standard form. 5 Ï }}} 2 2 4( )( ) }}} 2( ) T Use quadratic formula. 5 Simplify. Lengths cannot be, so use the solution. So, 5 <, and RS <. heckpoint omplete the following eercise. 3. Use the figure at the right to find JK. L M 8 K J opyright Holt Mcougal. ll rights reserved. Lesson 10. Geometry Notetaking Guide 287
6 THEOREM 10.1: SEGMENTS OF SENTS N TNGENTS THEOREM If a secant segment and a tangent segment share an endpoint outside a circle, then the product of the E lengths of the secant segment and its eternal segment equals the square of the length of the tangent segment. E 2 5 E p E Eample 3 Find lengths using Theorem 10.1 Use the figure at the right to find RS. 14 R S 12 Q RQ 2 5 RS p RT Use Theorem p ( 1 12 ) Substitute Simplify Write in standard form Ï }}} ( 1 )( 219 ) }}} 2( 1 ) T Use quadratic formula Ï } 58 Simplify. Lengths cannot be negative, so use the positive solution. So, Ï } 58 < 9.23, and RS < heckpoint omplete the following eercise. 3. Use the figure at the right to find JK. JK Ï } 73 L M 8 K J opyright Holt Mcougal. ll rights reserved. Lesson 10. Geometry Notetaking Guide 287
7 Eample 4 Fountain You are standing at point, 45 feet from the Point State Park fountain in Pittsburgh, P. The distance from you to a point of tangency on the fountain is 105 feet. Find the distance between you and your friend at point. p 5 Use Theorem p 5 5 You are Solve a real-world problem Substitute. Solve for. feet from your friend. 105 ft 45 ft heckpoint omplete the following eercise. 4. In Eample 4, suppose } is a diameter of the fountain. Use the diagram below and Theorem 10.1 to find the radius of the fountain. 105 ft r r 45 ft Homework 288 Lesson 10. Geometry Notetaking Guide opyright Holt Mcougal. ll rights reserved.
8 Eample 4 Solve a real-world problem Fountain You are standing at point, 45 feet from the Point State Park fountain in Pittsburgh, P. The distance from you to a point of tangency on the fountain is 105 feet. Find the distance between you and your friend at point. p 5 2 Use Theorem p Substitute Solve for. You are 245 feet from your friend. 105 ft 45 ft heckpoint omplete the following eercise. 4. In Eample 4, suppose } is a diameter of the fountain. Use the diagram below and Theorem 10.1 to find the radius of the fountain. 105 ft r r 45 ft 100 ft Homework 288 Lesson 10. Geometry Notetaking Guide opyright Holt Mcougal. ll rights reserved.
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