lnscribed Angles and Tangent Lines

lnscribed Angles and Tangent Lines Objectives: • Know and use the properties of tangent lines • Perpendicular at the point of tangency • Two tangent segments from the same point are congruent • Know the vocabulary of INSCIBED ANGLES • Understand and use the property that the size of an inscribed angle is always ½ the size of its intercepted arc • Understand and use the property that opposite angles of inscribed quadrilaterals are supplementary • Understand and use the property that the size of an angle formed Ьу а tangent line and а chord is ½ the size of its arc • Understand and use the property that the measure of an angle f formed Ьу two lines that intersect outside а circle is half the difference of the measures of the intercepted arcs. / \ . • Understand and use the property that for а given point and circle, ,, the product of the lengths of the two segments from the point to \ the circle is constant along any line through the point and the circle. The diagram and formula below show that the products of the secants and their outer segments are equal Tangent Lines ---•- � А tangent line is а line that intersects а circle at exactly 1 point Tangent Line Point of Tangency � lf а° line is tangent to а circle, then it is perpendicular (forms а 90 Angle) to а radius at the point of tangency. (' МА '(_е, � \.Т .А \V"l- Ь.. Examples --1'� 1) Find the value of х if, ED is tange�to circle С , ___,.,, ,, ... .__ --- , � l_ х·-ь2.. 0 -t п _ .,.........,... \ Е ___, • -- -·· 7 / _/ 2) Find the value of х if MIJ and MN are ta�gent fo circle О ___ [_ ,\-/ L ,и QVA,D\)...\�Q..Д L Lt- :::= t>r.DВs po1-y6CJ,vS х- Lмt.Jo == �G,o "' 3 и, о - l t'=l- - С/ D q {) -5 ··-f З) lf сн is tarч�ent\o circlc А, solvc for Нн� l<�tч;t�, of ;t? rounded to the n<�arest t(�ntl-,. с л х� .::- 1J 1 ,,� ri1 4) lf PQ is tangent to circle О, solve for х. р .. . .. -- 1 �, с. ...-..,,_ -.-: ��с, ( \.... 5) it 1s tricky! Q х -� " � HЧf6Гt:N�йG -=-S --к+f-:.. , 1 s )_ - х z.-._ .2 ( '•. "':=: G. { ',. , -Tangent Line Properties у.?· -t 1 � � /+ l'I-. +-'\) .,_ 7.:l� ;;:; )(' ➔• 1� � + �\ t'ltf -;: /� х • х--:-б " J • lf а line outside the circle is perp<�ndicular to а raditJS, then it is tangent • Show that there is а 90 ° angle forrned arнi you t,ave J right triangle Ьу using Н)е PyH,agoretio Ttн�ore111 • Example: Is М/., tang�nt to circle N? ·tl\t',{'.1 i > t/·-HI L, • 1 Н \ ( \ \ \ \:\ t • ,\ t-. / :.) \',., - [ ,\ �)(..{. �: 1 . 1 ' •! 1• 1 Example 6) is Р(/ .._ ��о circie О? PQ = 24, QO - • _,/ -• • - -- - -- о '1-Ч р \ lA��� \� "i)o i:=> 7) ls PQ 10, РО = 26 -z.,. '2tл� � fo -- с,,, � QO РО = 15 Tangent Line Properties ,,/ , \ • lf 2 segments start at the same point outside the circle and both are tangent to the circle, then_ the 2 segments are �--··· . :-.- . -c:::r congruent • lnscribed: Shape drawn inside circle • Circumscribed: Shape drawn outside circle 1 1 1 . PQ 1 1 4- "'2-А; '----s' �G ; s -т::., ,� �� k,'tangent to circle О? = 4, = 14, • \ -'� t: _ t о?,, с; = Example perimeter of 8) Circle О is inscribed in triangle АВС. Find the еаг to Ье tangent triangle АВС. Assume that all sides that арр to the circle are tangent. �&.,\ �.uе...'Тёд.,., о \?,- = lo � -z{_\_o)-\- \С>+ t &';) +\С;) 2U. � t + <ь9 + i 2-(� --- ---. Li�\9\"_ �_j ... ..... ,- that 9) Find the perimeter of the circumscribed triangle. AII sides арреаг to Ье tangent to the circle are tangent. \б Sin 10 // One More Thing '- • Remember, if а line intersects а circle in exactly one point it is called а TANGENT LINE • What if the line intersects а circle in exactly two points? What is it called? -1'\.. �c....д.rv-f L--{ ,-.) ь Е 1s А 11 lnscribed Angles An ins�rj!Jed __0J1gle is any angle formed Ьу 2 chords of а circle. This is NOT а central angle. ,-\ The size of an inscribed angle is always ½ the size of \ its intercepted агс \ � c.u.,, t>� D .д � 6 t... ь съ "-1'""� :L Р-. � С, L� и в Example -------------, 10} Solve for the missing variaЫe Remember: Аге➔ lnscribed Angle: f \ 1 Divide Ьу 2 _ __ "--- А / >(:::: 60 ° '- 11) Remember: lnscribed Angle ➔ Аге: Multiply Ьу 2 \) Properties of lnscribed Angles • lf two inscribed angles share the same intercepted arc, then they are congruent. 5д 1/V\ G J н� f2LЪ \>1--е .v р.. Си-С- LA.:::- L6 Angle А = Angle В Example 12,13} Solve for the missing variaЫes у Properties of lnscribed Angles • lf an inscribed angle opens to а diameter, then it is а right angle • Why? 14) SЬlve for the missing angles in the triangle � .1/ л ·-..___ ------с Properties of lnscribed Angles • Opposite angles of �scrib�d quadrilaterals are supplementary ==--,;,,,._-- __ 'i:,.. А+С=180 B+D= 180 Example -------------16) Solve for the missing�_�gles --- х� /jo -/О? -­ /�() -- с5 () ДLL. * o ve for the missing angles 17) S l Ч- L--S А --- в 90• 70• Properties of lnscribed Angles (• The si ze of an angle formed Ьу а tangent line and а chord is ½ the size of its arc u -v ��,--- 13) Solve for the missing variaЫe в Property of lntercepted arcs • The measure of ап angle formed Ьу two lines that intersect outside а circle is half the difference of the measures of the intercepted arcs. _______ _ с_ L \ -=- 8\ � ,l.QL - L\1i u: /..�С... -:-----------­ ( mL1=¾(x-y) ] 19) Find the value of z: 10 Property of secants I \ / / '-- • For а given point and circle, the product of the lengths of the two segments from the point to the circle is constant along any line through the point and the circle. The diagram and formula below show that the products of the secants and their outer segments are equal. \ • 20) Find the value of с using the given chord and secant lengtt,s in the diagram shown. 9 t)иr� 1!)&- = 4 \�с�-\q G "-'Т 1{2,.� ;:;. <='J +- l 'S -::: ovr-s 1 0с:: � >.?> f:::.� 1.е:"С, ..;; У-.+, .3 t" 2.., L-z.�')-== 1;� 7-Ч3 ::: IЗх -t- l�q \Ъ)(. = -=t,Ч i-.:_·· �ч7г·; J Г-------------------