G.C.2 STUDENT NOTES WS #5 1 Chord and Central Angle Properties Theorem – In the same circle or in congruent circles, if two chords are congruent then their minor arcs are also congruent. B If…. C AB = CD AB ≅ CD A Converse of the Theorem – In the same circle or in congruent circles, if two minor arcs are congruent, then the chords are congruent. B If….. C Then…. E D AB ≅ CD AB ≅ CD A Then…. E D mAB = mCD Proof of Theorem mAB = mCD AB = CD AB ≅ CD Proof of the Converse of the Theorem Given: AB ≅ CD B B Given: AB ≅ CD C C A A EA ≅ EB ≅ EC ≅ ED because they are all radii. These radii form two congruent ∆’s with the given by SSS. Thus the central angles ∠AEB ≅ ∠DEC establishing that the arcs that they subtend are also congruent, E D EA ≅ EB ≅ EC ≅ ED because they are all radii and ∠AEB ≅ ∠DEC because they are central angles on the congruent arcs. This creates two ∆’s that are congruent by SAS. The congruent ∆’s establish that AB ≅ CD . θ E θ D AB ≅ CD . EXAMPLES Find m CF = _____ Find x = ______ Find m CE = _______ F x C x E 8.5 cm 108° A C A x 8.5 cm A 10 cm E D 106° x = 360 - 106 – 106 – 33 = 115° F 2 cm D Equal arcs = Equal chords x = 10 cm x = 360 - 108 = 252/3 = 84° Theorem – If a diameter of a circle is perpendicular to a chord, then the diameter bisects the chord and its arc. Converse of the Theorem – If a chord is a perpendicular bisector of another chord, then the first chord is a diameter. E E If diameter BE is ⊥ to a chord, If BE is the perpendicular then AB ≅ BC ( AB = BC ) bisector of AC , then BE is a diameter. D And AF ≅ FC ( mAF = mFC ) A 2 cm 8.5 cm 33° B F C A B C G.C.2 STUDENT NOTES WS #5 2 Proof of the Theorem Proof of Converse of the Theorem E Given that If BE is a diameter that is perpendicular to a chord. DA ≅ DC because they are radii D and DB ≅ DB because they are common. ∆ADB ≅ ∆CDB by HL. C B F A The perpendicular bisector of AC must go through the center because it represents all points that are equidistant from A and C and the center is by definition equidistant to all points on the circle. Thus making AB ≅ BC and central ∠’s ∠ADB ≅ ∠CDB establishing congruent arcs. EXAMPLES Find x = _____ Find x = _____ (2 dec.) Find x = ______ x x 13 cm A A 4 cm x 4 cm 10 cm 5 cm 3 cm 1 cm 52 + y2 = 132 y= 12 cm d = 12 + 1 = 13 cm x = 6.5 cm 42 + 52 = x2 x ≈ 6.40 cm Theorem – If two chords are equidistant from the center, then they are congruent. If GE ≅ GF and A Converse of the Theorem – In the same circle, or in congruent circles, if two chords are congruent, then they are equidistant from the center. If AB ≅ CD A GE ⊥ AB and E E G B F D C GF ⊥ CD , then AB ≅ CD A G B F C Then G B F D GE ≅ GF C Proof of the Theorem E 42 + 32 = x2 x ≈ 5 cm D The 4 ≅ radii and the given help us to establish that ∆AEG ≅ ∆BEG ≅ ∆CFG ≅ ∆DFG by HL. The congruent ∆’s provide congruent sides and so AB ≅ CD (AB = CD). Proof of the Converse of the Theorem The 4 ≅ radii and the given A help us establish congruent E G ∆’s by SSS. The two congruent isosceles ∆’s must have ≅ B F D heights. Thus they are the C same distance, GE ≅ GF . G.C.2 STUDENT NOTES WS #5 3 EXAMPLES Find x = _______ Find x = _______ (E) Find x = _______ 6.5 cm x A A 12 2 x x A 4 cm 90° x = 2(6.5) = 13 cm 45 – 45 - 90 x = 4 2 cm 45 – 45 - 90 x = 12 cm G.C.2 WORKSHEET #5 Name: ________________________________ Period ______ 1. Determine the requested value. a) b) IJ = 10 cm, HA = 11 cm I c) 1 d) 78° 55° x H 220° 25° J x 140° 9 cm 91° 10 cm x 60° B 27° 11 cm 12 cm A 70° 113° K x = ___________ e) AK = ___________ f) x = ___________ h) x x = ___________ i) 3.5 cm 88° x x 74° x = ___________ x = ___________ 2. Determine the length of radius AC . a) b) x = ___________ x = ___________ c) d) AC = __________ (2 dec.) AC = __________ (E) c) d) C A 5 cm 24 cm AC = __________ B AC = __________ (2 dec.) 3. Determine the requested value. a) b) 20 cm 5 cm x = ___________ x = ___________ x = ___________ (E) x x = ___________ (E) G.C.2 WORKSHEET #5 2 4. Determine the requested value. Some of these questions are quite… “SPECIAL”. a) b) c) d) x = ______________ (E) x = ______________ (E) x = ______________ (E) x = ______________ (E) e) f) g) h) x = ______________ x = ______________ B C A AB = ______________ (E) x x = ______________ 5. A student questions the teacher…. “It makes sense that if you have congruent chords you would have congruent arcs but can you prove it?” Help the teacher by proving this to be true. D C E B A 6. Why would the perpendicular bisector of a chord have to be a diameter? 7. An ancient plate from the Mayan time period was drop at a museum. The curator wanted to put it back together but needed to find the center of the place to reference the restoration. If the largest piece looked like this… how could they find the center of the plate? (o uo^", 6.C.2 WORKSHEET #5 I I Y period 1. Determine the requested value. b) lJ = 10 cm, HA = 11cm a) -0 x= lLc^^ x= e) x= h) Yv x= 1c^ i) q,v r bL-O *4 x= I cu x= 2. Determine the tength of radius AC a) x= . b) A 4' AC= L,s?+ 13 c't" AC= (2 dec.) AC = 3 ,to cm f=*- (2 dec.) AC= (E) 3. Determine the requested value. *= 2( (A x= x= 66 crx (E) (E) G.C.2 WORKSHEET #5 2 4. Determine the requested value. Some of these questions are quite... ,SPEC|AL". x= sG cA (E) f) e) AB (E) x= x= 11"^ q^ x= (E) h) 8) = [email protected] Crr (E) x= x= (E) lL cw x= 5. A student questions the teacher.... "!t makes sense that if you have congruent chords you would have congruent arcs but can you prove it?,, Help the teacher by proving this to be true. Dvo.,: in red:i , ; ,6 , eu ,* A Ctp U A ffg, €s) /- Cev E LD7A P e-ca^s*- AZ.A ,/ v*Y gou"\te''\ 6s,'n1v "-artjLn-s -'z ,*ln:* 5. why would the perpendicular bisector of a chord have to be a diameter? 7. An ancient plate from the Mayan time period was drop at a museum. The curator wanted to put it back together but needed to find the center of the place to reference the restoration. lf the largest piece looked like this... how could they find the center of the plate? &YL), !.)^ ( (

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