Special Segments in a Circle

Transcription

1 pecial egments in a ircle Find measures of segments that intersect in the interior of a circle. Find measures of segments that intersect in the eterior of a circle. are lengths of intersecting chords related? he star is inscribed in a circle. It was formed b intersecting chords. egments and are two of those chords. When two chords intersect, four smaller segments are defined. F GMN INRING INI IRL In Lesson 10-, ou learned how to find lengths of parts of a chord that is intersected b the perpendicular diameter. ut how do ou find lengths for other intersecting chords? Intersecting hords Make Model raw a circle and two intersecting chords. R Name the chords Q and R intersecting at. raw and R Q. Q nalze 1. Name pairs of congruent angles. plain our reasoning.. How are and RQ related? Wh?. Make a conjecture about the relationship of, Q, R, and. he results of the activit suggest a proof for heorem heorem 10.1 If two chords intersect in a circle, then the products of the measures of the segments of the chords are equal. ample: You will prove heorem 10.1 in ercise 1. ample 1 Find. 1 Intersection of wo hords ubstitution Multipl. ivide each side b. Lesson 10-7 pecial egments in a ircle

2 Intersecting chords can also be used to measure arcs. ample olve roblems UNNL unnels are constructed to allow roadwas to pass through mountains. What is the radius of the circle containing the arc if the opening is not a semicircle? raw a model using a circle. Let represent the unknown measure of the segment of diameter. Use the products of the lengths of the intersecting chords to find the length of the diameter. 1 ft 8 ft 1 egment products ubstitution 8 ivide each side b 1. egment ddition ostulate or 0 ince the diameter is 0, r 0. ubstitution and addition GMN INRING OUI IRL Nonparallel chords of a circle can be etended to form secants that intersect in the eterior of a circle. he special relationship among secant segments ecludes the chord. tud ip Helping You Remember o remember this concept, the wording of heorem 10.1 can be simplified b saing that each side of the equation is the product of the eterior part and the whole segment. heorem 10.1 If two secant segments are drawn to a circle from an eterior point, then the product of the measures of one secant segment and its eternal secant segment is equal to the product of the measures of the other secant segment and its eternal secant segment. ample: You will prove this theorem in ercise 0. ample Intersection of wo ecants Find R if Q 1, QR, and. Let R. 1 Q R QR R R R ecant egment roducts (1 ) ( ) ubstitution 8 istributive ropert 0 8 ubtract 8 from each side. 0 ( 7)( ) Factor isregard negative value. 70 hapter 10 ircles

3 he same secant segment product can be used with a secant segment and a tangent. In this case, the tangent is both the eterior part and the whole segment. his is stated in heorem heorem If a tangent segment and a secant segment are drawn to a circle from an eterior point, then the square of the measure of the tangent segment is equal to the product of the measures of the secant segment and its eternal secant segment. W Z X Y ample: WX WX WZ WY You will prove this theorem in ercise 1. ample Intersection of a ecant and a angent. Find. ssume that segments that appear to be tangent are tangent. () ( ) 1 ( ) his epression is not factorable. Use the Quadratic Formula. b b ac a 1 1 (1)( 8 ) (1) Quadratic Formula a 1, b 1, c or isregard the negative solution..7 Use a calculator. oncept heck 1. how how the products for secant segments are similar to the products for a tangent and a secant segment.. FIN H RROR eck and Latisha are writing products to find. Who is correct? plain our reasoning. eck Latisha 8 = 8 = 8 8 = = ( + 8) = = = ( + )( - 1) = 1 Lesson 10-7 pecial egments in a ircle 71

4 . ON N raw a circle with two secant segments and one tangent segment that intersect at the same point. Guided ractice Find. Round to the nearest tenth if necessar. ssume that segments that appear to be tangent are tangent pplication 7. HIORY he Roman oliseum has man entrances in the shape of a door with an arched top. he ratio of the arch width to the arch height is 7:. Find the ratio of the arch width to the radius of the circle that contains the arch. 7 ractice and ppl For ercises , 7 ee amples 1 tra ractice ee page 77. Find. Round to the nearest tenth if necessar. ssume that segments that appear to be tangent are tangent KNO If ou remove a knob from a kitchen appliance, ou ma notice that the hole is not completel round. uppose the flat edge is millimeters long and the distance from the curved edge to the flat edge is about. millimeters. Find the radius of the circle containing the hole. 7 hapter 10 ircles oug Martin

5 1. ROOF op and complete the proof of heorem Given: W Y and Z X intersect at. rove: W Y Z X W Z X Y tatements Reasons a. W Z, X Y a.? b.? b. imilarit c. W X c.? Z Y d.? d. ross products Find each variable. Round to the nearest tenth, if necessar L 8. G O M 10 onstruction Worker onstruction workers must know how to measure and fit shapes together to make a sound building that will last for ears to come. hese workers also must master using machines to cut wood and metal to certain specifications that are based on geometr. Online Research For more information about a career as a construction worker, visit: com/careers. ONRUION n arch over a courtroom door is 0 centimeters high and 00 centimeters wide. Find the radius of the circle containing the arc of the arch. 0. ROOF Write a two-column 1. ROOF Write a two-column proof proof of heorem of heorem Given: secants and rove:. RIIL HINKING In the figure, Y is the midpoint of X Z. Find WX in terms of XY. plain our reasoning. Z Given: tangent R, secant U rove: (R) U W Y X 00 cm Lesson 10-7 pecial egments in a ircle 7 R U 0 cm avid Young-Wolff/hotodit

6 tandardized est ractice. WRIING IN MH nswer the question that was posed at the beginning of the lesson. How are the lengths of intersecting chords related? Include the following in our answer: the segments formed b intersecting segments, and, and the relationship among these segments.. Find two possible values for from the information in the figure.,,,, ft (0 ) ft. LGR Mr. Rodriguez can wash his car in 1 minutes, while his son Marcus takes twice as long to do the same job. If the work together, how long will it take them to wash the car? min 7. min 10 min. min Maintain Your kills Mied Review Find the measure of each numbered angle. ssume that segments that appear tangent are tangent. (Lesson 10-) Find. ssume that segments that appear to be tangent are tangent. (Lesson 10-) INIR MURMN Joseph lackarrow is measuring the width of a stream on his land to build a bridge over it. He picks out a rock across the stream as landmark and places a stone on his side as point. hen he measures feet at a right angle from and marks this. From, he sights a line to point on the other side of the stream and measures the angle to be about 7. How far is it across the stream rounded to the nearest whole foot? (Lesson 7-) lassif each triangle b its sides and b its angles. (Lesson -1) M 1 K L 10. Getting Read for the Net Lesson 7 hapter 10 ircles RRQUII KILL Find the distance between each pair of points. (o review the istance Formula, see Lesson 1-.). (, 7), (10, 1) 7. (1, 7), F(, ) 8. G(, ), H(1, )