8.5 Circles and Lengths of Segments

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1 LenghofSegmen nb Cicle and Lengh of Segmen In hi ecion we will how (and in ome cae pove) ha lengh of chod, ecan, and angen ae elaed in ome nal way. We will look a hee heoem ha ae hee elaionhip involving podc of pa of chod, pa of ecan, and pa of angen. The poof will involve he e of imila iangle. êêêêê êêêêê êêêêê êêêêê In he fige below, chod AB and CD ineec inide ŸO. AM and MB ae called êêêêê êêêêê êêêêê êêêêê he egmen of chod AB ( he ame can be aid fo CM and MD fo chod CD). The phae "egmen of a chod" will efe o he lengh of a egmen a well a he egmen ielf (in he ame way ha we e he em "adi" and "diamee"). B C M O A D The heoem below ae he elaionhip beween he egmen of hee wo chod. Theoem: When wo chod ineec inide a cicle, he podc of he egmen of one chod eqal he podc of he egmen of he ohe chod.

2 LenghofSegmen nb 2 Remembe ha we ae implying "lengh of he egmen" in he heoem. The poof i hown below. B C M O A oof: D êêêêê êêêêê iven: AB and CD ineec a M. ove: = Saemen Reaon 1. Daw chod AD êêêêê and CB êêêê. 1. Thogh any wo p. hee i exacly one line. 2. C; B 2. If wo incibed angle inecep he ame ac, hen he ' 3. DAMD ~ DCMB 3. AA ~ olae 4. ÅÅÅ = ÅÅÅÅ 4. Co. ide of ~ D' ae in pop. 4. = 5. opey of popoion êêêê êêêê êêêê êêêê In he diagam fo he nex heoem, and ae ecan egmen. and H ae exeio o he cicle and ae efeed o a exenal egmen. The em "ecan egmen" and "exenal egmen" will be ed o efe o he lengh of a egmen a well a he egmen ielf. Look a he heoem below.

3 LenghofSegmen nb 3 Theoem: When wo ecan egmen ae dawn o a cicle fom an exenal poin, he podc of one ecan egmen and i exenal egmen eqal he podc of he ohe ecan egmen and i exenal egmen. iven: êêêê and êêêê dawn o he cicle fom poin. ove: = H oof: Saemen Reaon 1. Daw chod H êêêê êêêê and. 1. Thogh any wo p. hee i exacly one line. 2. If wo incibed angle inecep he ame ac, hen he ' 3. Reflexive opey 4. DH ~ D 4. AA ~ olae 5. ÅÅÅ = ÅÅÅÅ 5. Co. ide of ~ D' ae in pop. 6. = 6. opey of popoion The la heoem i eally j a pecial cae of he heoem above. We will e he ame line of eaoning hee a we did in he la ecion wih an angle fomed by a chod and a angen (which wa a pecial cae of an incibed angle). Look a he hee diagam below.

4 LenghofSegmen nb 4 H H = =H In he fi wo diagam i i eay o ee ha =. In he hid diagam and boh become eqal o he lengh of he angen egmen, and he elaionhip become = 2. Thi lead o he following heoem.

5 LenghofSegmen nb 5 Theoem: When a ecan egmen and a angen egmen ae dawn o a cicle fom an exenal poin, he podc of he ecan egmen and i exenal egmen i eqal o he qae of he angen egmen. iven: Secan egmen êêêê and angen egmen êêêê dawn o he cicle fom. ove: = 2 lan fo poof: Daw chod êêêê êêêê and. Show ha and ae congen becae hey inecep he ame ac. Then how ha D and D ae imila iangle and e he popeie of imila iangle and he popeie of popoion o finih he poof. In he following example chod, ecan, and angen ae hown. xample 1: ind he vale of x. 3 2 x 8

6 LenghofSegmen nb 6 Solion: Since we have wo chod ineecing in he ineio of a cicle we can apply he fi heoem of hi ecion. 2 x = 3 8 2x = 24 x = 12 xample 2: ind he vale of x. x Solion: Thi ime we ae dealing wih wo ecan egmen o we apply he econd heoem in hi ecion. 4 ( x + 4) = 5 ( 12 ) 4x + 16 = 60 4x = 44 x = 11

7 LenghofSegmen nb 7 xample 3: ind he vale of x. 10 3x x Solion: We ee a ecan egmen and a angen egmen o we apply he hid heoem of hi ecion. 4x ( x) = x 2 = 100 x 2 = 25 x = 5 We can ignoe he negaive oo ince we ae dealing wih egmen lengh.

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