GEOMETRY OF THE CIRCLE TANGENTS & SECANTS

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1 Geometry Of The ircle Tngents & Secnts GEOMETRY OF THE IRLE TNGENTS & SENTS

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3 Tngents TNGENTS nd N Secnts SENTS Tngents nd secnts re lines tht strt outside circle. Tngent touches the circle t one point nd Secnt cuts through circle. This section hs some theorems out how to use these lines to solve prolems. nswer these questions, efore working through the chpter. I used to think: Wht is the 'point of contct' of tngent to circle? Shde the lternte segment of the circle to + E E Wht does this men: "The products of the intercepts of two intersecting chords re equl"? nswer these questions, fter working through the chpter. ut now I think: Wht is the 'point of contct' of tngent to circle? Shde the lternte segment of the circle to + E E Wht does this men: "The products of the intercepts of two intersecting chords re equl"? Wht do I know now tht I didn t know efore? K 1 16 SERIES TOI

4 sics Secnts secnt psses through circle. The two points (on the circle) which the secnt psses through re clled 'intercepts'. Secnt Intercept Q R S Secnt Intercept In the ove digrm, the line is secnt which intercepts the circle t nd. S is secnt tht goes through the centre of the circle. S intercepts the circle t points Q nd R. Tngents tngent is stright line tht touches the circle once. This point is clled the point of contct. Every other point on the line lies outside the circle. F oint of contct E F is tngent in the ove digrm. E is the point of contct. 2 K 16 SERIES TOI

5 Questions sics 1. Wht is the difference etween secnt nd tngent? 2. rw secnt nd tngent to the circle elow. Lel the "point of contct". 3. Wht s the difference etween n intercept nd "point of contct"? K 3 16 SERIES TOI

6 Knowing More Tngents nd Secnts hve certin properties which we cn use to find ngles nd side lengths involving circles. Using Tngents to Solve rolems Here re some theorems showing how tngents cn e used with circles. Theorem 9: Tngents re perpendiculr to rdius t the point of contct O Given: O is the centre of the circle To rove: O = roof roof onstruct line O where is on the tngent t ny other point thn The shortest distnce etween point nd line is the perpendiculr distnce Since is on the circle, nd will lwys e outside the circle O will lwys e greter thn the rdius. ` O 1 O lwys. O is the shortest distnce nd this hs to e perpendiculr. This lso mens tht if line which touches the circle is perpendiculr to the rdius (or dimeter), then the line is tngent. Theorem 10: If two tngents re drwn to circle from common externl point, then they re equl O Given: O is the centre of the circle To rove: = roof roof onstruct O, nd rdii O nd O In TO nd TO + O = + O = 90c (Tngent = rdius) O = O (Equl rdii) O is common ` TO / TO (RHS) ` = (orresponding sides of congruent tringles) Two tngents drwn to circle hve equl length if they hve the sme source. 4 K 16 SERIES TOI

7 Knowing More Here re some exmples demonstrting how to use Theorem 9 nd Theorem 10 In the digrm elow, Q nd R re tngents to circle with centre O nd rdius 11.5 cm. O = 29.9cm Find the length of R OR = R (Tngent perpendiculr to rdius t point of contct) ` + OR = 90c ` R = O -OR ` R = = 276. (ythgors) Q R O Find the length of Q Q = R ` Q = 27.6 cm (Tngents from common source re equl) ircle elow hs its centre t O nd Q is the dimeter. Show tht if nd re oth tngents. = Q ` + O = 90c (Tngent perpendiculr to rdius t point of contct) O = Q ` + QO = 90c (Tngent perpendiculr to rdius t point of contct) Q ` (lternte ngles re equl) O is the centre of the circle elow. FG is tngent to the circle with point of contct N. Show tht E FG O M N E G E = ON ` + OME = 90c FG = ON ` + ONG = 90c (Line from centre to midpoint is perpendiculr to chord) (Tngent perpendiculr to rdius t point of contct) F ` E FG (orresponding ngles re equl) K 5 16 SERIES TOI

8 Questions Knowing More 1. O is the centre of the circle elow: U Show tht TTUO / TTVO (SSS) Show tht OT isects + UTV T O V 2. Find the length of tngent YZ if O is the centre of the circle. O 24cm 62. 4cm Y Z 6 K 16 SERIES TOI

9 Questions Knowing More 3. R is tngent to the circle with point of contct Q. O is the centre of the circle. + Q = 15c nd + RQ = 30c. Find reflex + O. Q R O 4. O is the centre of the circle elow. nd re tngents with points of contct S nd T respectively. Show tht + S = + TS. S O T K 7 16 SERIES TOI

10 Questions Knowing More 5. M is tngent to the lrger circle nd M is tngent to the smller circle. M is common tngent. O is the centre of the lrger circle. Find the rdius of the lrger circle if M = 19. 5cm nd MO = 22. 1cm. O M 8 K 16 SERIES TOI

11 Using Our Knowledge The lternte Segment In ech of the digrms elow, Q is tngent with point of contct. The lternte segment in reltion to + Qhs een shded. ngle in lternte segment ngle in lternte segment Q Q The 'lternte segment' to + Q is the segment which does not contin + Q. The ngle in the circle opposite + Q is clled the 'ngle in the lternte segment'. In the digrms ove, the ngle in the lternte segment is+ Which is the ngle in the lternte segment in reltion to nswer = + Identify the ngle in the lternte segment of T +? + TU nd + SU S ngle in lternte segment of ngle in lternte segment of + TU is + TSU + SU is + STU U Identify the ngle in the lternte segment of + HFN, + GFM nd + HFM H + FGH is in the lternte segment to + HFN G I + GHF is in the lternte segment to + GFM + HIF is in the lternte segment to + HFM M F N K 9 16 SERIES TOI

12 Using Our Knowledge Theorem 11: n ngle etween tngent nd chord t the point of contct is equl to the ngle in the lternte segment M Given: O is the centre of the circle. Q is tngent with point of contct To rove: + Q = + roof roof O onstruct imeter M nd join M Q M = Q ` + MQ = 90c + Q = 90c -+ M (Tngent perpendiculr to rdius t point of contct) now + M = 90c ` + M + + M + 90c = 180c ` + M = 90c -+ M ` + Q = + M (ngle in semicircle) (Sum of ngles in tringle) (oth equl 90c - + M ) now + = + M ` + Q = + (ngles sutended y sme rc in sme segment) (oth equl + M ) This is lso clled the lternte Segment Theorem. Here re some exmples how to use this theorem Find the size of X + LMN L + NLM = + MNY ` + NLM = 42c (lternte Segment ngle) N 73c 42c Y M + LMN+ + LNM + + NLM = 180c ` + LMN + 73c+ 42c = 180c ` + LMN = 65c (ngle sum of tringle) R is common tngent to the two circles leow. Find + Q nd + Q + Q = + QR = 68c + Q = + Q = 76c (lternte Segment Theorem) (lternte Segment Theorem) 76c Q 68c R Show + Q = + QR = ` + Q = + Q ` 68c (lternte Segment Theorem) (oth equl 68c ) (orresponding ngles re equl) 10 K 16 SERIES TOI

13 Questions Using Our Knowledge 1. Identify the ngles equl to the lelled ngles. eg c 2. Find the size of + LJI if IJ is tngent to the circle elow. K 71c L 44c I J K SERIES TOI

14 Questions Using Our Knowledge 3. E nd re oth tngents to the circle elow. Find + nd + FE. E F 68c 87c 4. The circle elow hs centre O nd tngent Q with point of contct F. Find + OE + OF c + EFQ O 30c E 50c F Q 12 K 16 SERIES TOI

15 Questions Using Our Knowledge 5. nd re tngents to the circle with points of contct nd S respectively. RS isects + QS. Show tht Find the size of T QS is n Isosceles tringle. + QRS. 70c 70c c Find the size of + RS. Q S R K SERIES TOI

16 Thinking More Lengths of Secnts, hords nd Tngents The next 3 theorems del with the lengths of secnts, chords nd tngents (insted of the sizes of ngles in circles) Theorem 12: The products of the lengths of the segments of two intersecting chords re equl Given: nd intersect ech other t E To rove: E # E = E # E roof E roof Join nd In T E nd TE + E = + E + E = + E + E = + E ` TE TE ` E = E E E ` E # E = E # E (Verticlly opposite ngles) (ngles in sme segment on sme rc) (ngles in sme segment on sme rc) (Equingulr) (Rtio of corresponding sides for similr T 's ) This mens tht if two chords cut ech other, the product of the lengths of the cut chords re equl. Find the length of TS (ll mesurements in cm) RT TS T TQ # # = (roducts of intercepts on intersecting chords) R 12 8 T 9 Q S ` 12 # TS = 8# 9 ` TS = = 6 cm Find the length of chord (ll mesurements in cm) 10 5 E 14 E # E = E # E ` 10 # E = 14 # 5 ` E = = 8 cm = E + E (roducts of intercepts on intersecting chords) = = 18 cm 14 K 16 SERIES TOI

17 Thinking More Theorem 13: The products of the intercepts of two intersecting secnts to circle from n externl point re equl S To rove: S S S S # # = roof roof Join nd In T S nd TS + S = + S + S = + S ` TS TS (ommon ngle) (ngles in sme segment on sme rc) (Equingulr) ` S = S S S ` S # S = S # S (orresponding sides of similr tringles) This mens tht if two secnts (to the circle) hve the sme strting point, the products of the entire secnt nd the distnce to the circle re equl. Find the length of E (ll mesurements in cm) H 40 G E F F EF HF GF # # = ` F # 50 = 100# 60 ` F = 6000 = 120 cm 50 E = F -EF = = 70 cm (roducts of intercepts of interesecting secnts from externl point) Find x in the digrm elow (ll mesurements in cm) x 8 E E x # # = ` ( x+ 8) x = 24# 10 ` x 2 + 8x- 240 = 0 ( x- 12)( x+ 20) = 0 (roducts of intercepts of interesecting secnts from externl point) ` x = 12 or x =-20 Since length is lwys positive: x 12 = K SERIES TOI

18 Thinking More Theorem 14: The squre of tngent to circle equls the product of the intercepts of secnt from the sme externl point Given: T is tngent with point of contct T 2 To rove: ( T) = # roof roof Join T nd T T In T T nd TT + T = + T + T = + T ` TT TT (ommon ngle) (lternte Segment Theorem) (Equingulr) ` T = T 2 ` ( T) = # (orresponding sides of similr tringles) This mens tht if tngent nd secnt hve the sme source, then the product of the intercepts of the secnt is equl to the squre of the tngent. Find the length of the tngent in the following digrm (ll mesurements in cm) 2 ( ) = # (Squre of tngent equl product of secnt from common point) 2 ` ( ) = 80 # 20 = ` = = 40 cm Find the length of x in the following digrm (ll mesurements in cm) R 10 Q x 2 ( S) = x# R 2 ` ( 12) = xx ( + 10) ` x x- 144 = 0 (Squre of tngent equl product of secnt from common point) S 12 ` ( x- 8)( x+ 18) = 0 ` x = 8 or x =-18 Since length is lwys positive: x 8 = 16 K 16 SERIES TOI

19 Questions Thinking More 1. Find x in ech of the following (ll mesurements in cm) E x V 9 U 5 3 x T c Given: Q is tngent S Q 16 x 9 R K SERIES TOI

20 Questions Thinking More 2. Find the missing lengths in ech of the following (ll mesurements in cm). E E = 8 = 7 = 5 Find = x L M K N KL = 12 MN = 8 Find LM = x 18 K 16 SERIES TOI

21 Questions Thinking More 3. Find x nd y in the digrm elow: 8 y x E K SERIES TOI

22 = Tngents nd Secnts Visul Theorems Visul Theorems Here is visul summry of ll the theorems in this ooklet. Theorem 9: Tngents re perpendiculr to rdius t the point of contct Theorem 12: The products of the intercepts of two intersecting chords re equl Rdius c d c d # # = Tngent Theorem 10: If two tngents re drwn to circle from common externl point, then they re equl Theorem 13: The products of the intercepts of two intersecting secnts to circle from n externl point re equl = d c c d # # = Theorem 11: n ngle etween tngent nd chord is equl to the ngle in the lternte segment Theorem 14: The squre of tngent to circle equls the product of the intercepts of secnt from the sme externl point t t 2 # = 20 K 16 SERIES TOI

23 Questions Thinking Even More Here is mix of more difficult prolems comining ll the theorems for ircle Geometry. 1. O is the centre of the circle elow. Q is tngent with point of contct. Q 30c + =. Find 5 other ngles which equl 30c. O = = 30c Q K SERIES TOI

24 Questions Thinking Even More 2. E is Rhomus, G is tngent to the circle t E nd is stright line. Show + GE = + E. = c Show E isects + E. Show + E+ + E = 180c. = = = G E 22 K 16 SERIES TOI

25 Questions Thinking Even More 3. In the digrm elow, O is the centre of the circle nd J is the point of contct of tngent KJ. Given JK = cm J = cm O = OM = 765. cm + NO = 37c O = NJ nd OM = NL J Find the length of N. Find the length of LN. N 37c - O M L K c d Find the length of LK. Find + JOL. K SERIES TOI

26 Questions Thinking Even More 4. JM nd LM re tngents with points of contct J nd L respectively. JK ML nd KM is stright line. Show + LJK = + LJM. Show + JML = + LKJ. L K J N M 24 K 16 SERIES TOI

27 Questions Thinking Even More 5. In the digrm elow QR nd Q re tngents with points of contct R nd respectively. Let+ QW = x; + OV = y nd + WQR = z. Show + TQ = + RQ. Q z x y O Show TQ ;; UR. V c Show + VW = + URT = 180c. d Show + URS = + WQR. W T R U S K SERIES TOI

28 Tngents & Secnts nswers sics: 1. secnt psses through circle. The two 1. points it psses through re clled intercepts. While tngent is stright line tht touches the circle once. This point is clled the point of contct. Using Our Knowledge: 2. c 3. point of contct touches the circle once. n intercept goes through the the circle LJI = 65c 2. Knowing More: YZ = 57.6 cm 3. + = 87c + FE = 25c 3. + O = 270c 4. + OE = 20c 5. O = cm + OF = 40c c + EFQ = 60c 5. + QRS = 110c c 145c 26 K 16 SERIES TOI

29 Tngents & Secnts nswers Thinking More: 1. x 30 cm = x = 4 cm c x = 15 cm 2. x = 6 cm x = 4 cm 3. x = 6 cm y = cm Thinking Even More: 1. + = 30c + = 30c + = 30c + = 30c + = 30c 3. c d N = 10.2 cm LN = 20.4 cm LK = 6.8 cm + JOL = 148c K SERIES TOI

30 Notes 28 K 16 SERIES TOI

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