Eam aer 3 ircle Geomery (ar 3) emen andard:.4.(c) yclic uadrilaeral La week we covered u otheorem 3, he idea of a convere and we alied our heory o ome roblem called IE. Okay, o now ono he ne chunk of heory which ha o do wih YLI UILTEL. Now remember - a uadrilaeral i any four ided figure. yclic uadrilaeral i a four ided figure wih all four corner lying on he circumference of a circle. We uually name i clockwie or ani-clockwie -don ZIGZG. ollec your aer 3 Leon every week!! Guy, boh N and IE eaminaion candidae have he oion of wriing aer 3 a he end of he year! aer 3 cover addiional mahemaic maerial and i ou of 00 mark. Mah aer 3 will really e you aar in he job marke, and make udying echnical ubjec a eriary level eaier. We have hooked you u wih hee leon - wrien by IE Mah aer 3 eaminer, Heaher Frankiko. The Eaminaion Guideline for he OE iulae ha he roof of all Grade and he Grade heorem of he geomery of circle are required for eaminaion uroe. Thi bookwork ha a uggeed maimum mark allocaion of 5 ou of 00. The IE ubjec emen Guideline ugge ha no bookwork i.e. NO OOF be eamined. We can eec omlee he aemen ye queion bu no comlee roof. Thee Geomery module will no reen roof. You mu ge hee from your eacher. W T yclic quad abcd/dabc/cbad yclic quad W No cyclic T No cyclic No Theorem 5: The ooie angle of a cyclic quadrilaeral are ulemenary (ooie of a cyclic quad) and converely, if a air of ooie angle of a quadrilaeral i ulemenary hen he quad i cyclic. (Ooie of quad ul.) age
Eam aer 3 ircle Geomery (ar 3) If cyclic hen + = 80º + = 80º If + = 80º hen cyclic + = 80º 80 - Theorem 6: The eerior angle of a cyclic quad i equal o he inerior ooie angle (e of cyclic quad) and converely, if he eerior angle of a quad i equal o he inerior ooie angle, he quad i cyclic (e of quad) If cyclic hen = 80º If = hen i cyclic (we can draw a circle hrough all 4 oin) 80º Le u ee if you can aly hee YLI U Theorem here. age 3
Eam aer 3 ircle Geomery (ar 3) E an you ee he following? K 3 G F ) E = ( ame egmen) F = ( ame egmen) ) F + = 80º (o cyclic quad EF) + = 80º (o cyclic quad E) E + + = 80º (o cyclic quad E) E + E + = 80º (o cyclic quad EF) 3) = F (e cyclic quad EF) 3 = E (e cyclic quad E) F = (e cyclic quad FE) LEE look a hi IE carefully - ue your finger or colour (or whaever) o ry o ee all he angle menioned above. lo remember ha you have me an eerior angle before. o you remember ha for any riangle, he eerior angle (formed by eending any ide) wa equal o he um of he inerior ooie wo angle? o = + Here for a cyclic quadrilaeral he eerior angle i equal o he one inerior ooie angle. K = Noe: bf mu be raigh like K F age 4
Eam aer 3 ircle Geomery (ar 3) The ne bi of heory relae o a very ecial line called a TNGENT We define a TNGENT o be a line which make conac wih a circle a only one oin on he circumference. In each ider below T i a angen. In each of he above keche, draw in he radiu O. an you ee ha O will alway be erendicular o T? Thi i your ne heorem. Theorem 7: angen o a circle i alway erendicular o he radiu a he oin of conac (rad angen) and converely, a line drawn erendicular o a radiu a he oin where i mee he circumference, will be a angen (radiu o line) Given T a angen wih O a radiu hen O T and converely, Given radiu O and T O hen T i a angen age 5
Eam aer 3 ircle Geomery (ar 3) Theorem 8: If wo angen are drawn o a circle from a common eernal oin, hen he wo diance from hi oin o he oin of conac are equal (ang, ame oin) = q (boh mu be angen from ) i acually called a ecan nd now for he final, bu he mo ued heorem in ircle Geomery. We call i he TN-HO heorem Thi heorem ay Theorem 9: The angle formed beween a angen o a circle and a chord drawn from he oin of conac... chord drawn a oin of conac angle beween angen and chord angen o circle i equal o he angle ubended by he ame chord in he oher egmen (an chord heorem) i he angle beween angen and chord M coming in from he righ y m y i he angle beween angen and chord coming in from he lef y T age 6
Eam aer 3 ircle Geomery (ar 3) 3 4 W In he diagram T and W are angen an you agree o all hee aemen? ) T = (an chord T) T 4 = 3 = T 4 = T 3 (an chord TK) (an chord ) (an chord K) 3 4 K ) T + T = K (an chord T) T 3 + T 4 = b (an chord T) + = K (an chord T) 3 + 4 = b (an chord T) 3) T = (angen ame T) \ T 3 + T 4 = 3 + 4 (Io ) 4) + K = 80 (o cyclic quad) + 3 + T + T 3 = 80 (o cyclic quad) 5) There are no eerior angle of cyclic quad TK in he diagram 6) There i no radiu / diameer/ cenre o no angle i 90 Of coure hi very imoran Theorem mu have a convere... If he angle formed beween a chord and a line i equal o an angle ubended by he chord, hen he line i a angen o he circle (convere an chord) c If = hen we can aume ha a circle could be drawn going hrough, and Thi would make a chord and he oin of conac wih angen T o now you have all he heory maered. Le u ry an eam ye rider miing i all u. age 7
Eam aer 3 ircle Geomery (ar 3) emember o read he given informaion lowly; ay aenion o keyword uch a cenre, radiu, cyclic quad, angen and relae hem o heir heorem. Mark all informaion on icure. We will ar wih an eay one ider 65 3 Given:,, and are oin on a circle. i he diameer of he circle MT i a angen a = 65 and 3 = 35 M 35 3 4 T eermine, wih reaon, he ize of ; 4 and 3 Think: mu be cyclic / i a diameer o + = 90 / mu be o T he angen oluion a) = (an chord ) \ = 65 b) 3 + 4 = 90 (rad ang) \ 4 = 55 c) + = 90 ( in emi-circle) \ 3 = 90 ( on line) You were no aked o find any oher angle bu ee if you ge = 65 ; = 5 ; = 5 ; T = 35 ; = 35 ec. age 8