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5. AB
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CHAPTER 6 Circles
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6.18 Proving an angle measurement problem stated in words Prove that parallel chords drawn at the ends of a diameter are equal in length.
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Given: Circle O
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AB is a diameter. AC i BD
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To Prove: Plan:
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PROOF:
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Prove AC > BD
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Statements 1. 2. 3. 4. 5. AB is a diameter ACB > ADB AC i BD AD > BC AC > BD BD 1. 2. 3. 4. 5.
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Reasons Given A diameter cuts a circle into two equal semicircles. Given Parallel lines intercept > arcs on a circle. If equals are subtracted from equals, the differences are equal. Definition of > arcs. 6. In a circle, equal arcs have chords which are equal in length.
6. AC
SUPPLEMENTARY PROBLEMS
6.1. Provide the proofs requested in Fig. 6-44. (a) Given: AB DE AC DF To Prove: jB > jE (b) Given: Circle O, AB BC Diameter BD To Prove: BD bisects jAOC. (c) Given: (6.3) Circle O AB CD To Prove: jAOC > jBOD
(b) Fig. 6-44
Provide the proofs requested in Fig. 6-45. Please refer to figure 6-45(a) for problems 6.2(a) and (b); to figure 6-45(b) for problems 6.2(c) and (d); and figure 6-45(c) for problems 6.2(e) and (f). (6.3) (a) Given: AB AC To Prove: ABC > ACB (b) Given: ABC > ACB To Prove: AB AC (c) Given: Circle O, AB AD Diameter AC To Prove: BC CD (d) Given: Circle O AB AD, BC CD To Prove: AC is a diameter. (e) Given: AD BC To Prove: AC BD (f) Given: AC BD To Prove: AD BC
CHAPTER 6 Circles
Fig. 6-45
Prove each of the following: (a) If a radius bisects a chord, then it bisects its arcs. (b) If a diameter bisects the major arc of a chord, then it is perpendicular to the chord. (c) If a diameter is perpendicular to a chord, it bisects the chord and its arcs.
(6.4)
Prove each of the following: (a) A radius through the point of intersection of two congruent chords bisects an angle formed by them.
(6.4)
(b) If chords drawn from the ends of a diameter make congruent angles with the diameter, the chords are congruent. (c) In a circle, congruent chords are equally distant from the center of the circle. (d) In a circle, chords that are equally distant from the center are congruent. 6.5. Determine each of the following, assuming t, t9, and t99 in Fig. 6-46 are tangents. (a) If mjA (b) If BR 90 in Fig. 6-46(a), what kind of quadrilateral is PAQO RC in Fig. 6-46(b), what kind of triangle is ABC (6.5)
(c) What kind of quadrilateral is PABQ in Fig. 6-46(c) if PQ is a diameter (d) What kind of triangle is AOB in Fig. 6-46(c)
Fig. 6-46
In circle O, radii OA and OB are drawn to the points of tangency of PA and PB. Find mjAOB if mjAPB equals (a) 40 ; (b) 120 ; (c) 90 ; (d) x ; (e) (180 x) ; (f) (90 x) . (6.6) Find each of the following (t and t9 in Fig. 6-47 are tangents). In Fig. 6-47(a) (a) If mjPOQ (b) If mjPBO (c) If mjPAQ 80 , find mjPAQ. 25 , find mj1 and mjPAQ. 72 , find mj1 and mjPBO. (6.6)
CHAPTER 6 Circles
Fig. 6-47
In Fig. 6-47(b) (d) If PB bisects jAPQ, find mj2. (e) If m j1 (f) If PQ 6.8. 35 , find mj2. QB, find mj1. (6.7)
In Fig. 6-48(a), ^ABC is circumscribed. (a) If y 9, find x. (b) If x 25, find y. In Fig. 6-48(b), quadrilateral ABCD is circumscribed. (c) Find AB CD. (d) Find perimeter of ABCD. In Fig. 6-48(c), quadrilateral ABCD is circumscribed. (e) If r 10, find x. (f) If x 25, find r.
Fig. 6-48
If two circles have radii of 20 and 13, respectively, find their line of centers: (a) If the circles are concentric (b) If the circles are 7 units apart (c) If the circles are tangent externally (d) If the circles are tangent internally
(6.8)
If the line of centers of two circles measures 30, what is the relation between the two circles: (a) If their radii are 25 and 5 (b) If their radii are 35 and 5 (c) If their radii are 20 and 5 (d) If their radii are 25 and 10
(6.8)
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