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s 6. Corresponding parts of congruent ^ are >.
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6.4 Proving a circle problem stated in words Prove that if a radius bisects a chord, then it is perpendicular to the chord.
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Given: Circle O OC bisects AB.
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To Prove: OC ' AB Plan: Prove ^ AOD > ^ BOD to show j 1 > j 2. Also, j 1 and j 2 are supplementary.
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Statements 1. Draw OA and OB. 2. 3. 4. 5. 6. 7. 8. OA > OB OC bisects AB. AD > DB OD > OD ^ AOD > ^ BOD j1 > j2 j 1 is the supplement of j 2.
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Reasons 1. A straight line segment may be drawn between any two points. 2. Radii of a circle are congruent. 3. Given 4. To bisect is to divide into two > parts. 5. Reflexive property 6. SSS s 7. Corresponding parts of congruent ^ are >. 8. Adjacent are supplementary if exterior sides lie in a straight line. 9. Congruent supplementary angles are right angles. 10. Rt. are formed by perpendiculars.
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9. j 1 and j 2 are right angles. 10. OC ' AB.
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CHAPTER 6 Circles
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6.2 Tangents
The length of a tangent from a point to a circle is the length of the segment of the tangent from the given point to the point of tangency. Thus, PA is the length of the tangent from P to circle O in Fig. 6-16.
Fig. 6-16
6.2A Tangent Principles
PRINCIPLE
A tangent is perpendicular to the radius drawn to the point of contact.
Thus if AB is a tangent to circle O at P in Fig. 6-17, and OP is drawn, then AB ' OP.
PRINCIPLE
A line is tangent to a circle if it is perpendicular to a radius at its outer end.
Thus if AB radius OP at P of Fig. 6-17, then AB is tangent to circle O.
Fig. 6-17
PRINCIPLE
Fig. 6-18
A line passes through the center of a circle if it is perpendicular to a tangent at its point of contact.
Thus if AB is tangent to circle O at P in Fig. 6-18, and CP ' AB at P, then CP extended will pass through the center O.
PRINCIPLE
Tangents to a circle from an outside point are congruent.
Thus if AP and AQ are tangent to circle O at P and Q (Fig. 6-19), then AP > AQ.
PRINCIPLE 5:
The segment from the center of a circle to an outside point bisects the angle between the tangents from the point to the circle.
Thus OA bisects j PAQ in Fig. 6-19 if AP and AQ are tangents to circle O.
Fig. 6-19
CHAPTER 6 Circles
6.2B Two Circles in Varying Relative Positions
The line of centers of two circles is the line joining their centers. Thus, OO9 is the line of centers of circles O and O9 in Fig. 6-20.
Fig. 6-20
Fig. 6-21
Circles Tangent Externally Circles O and O9 in Fig. 6-21 are tangent externally at P. AB 4 the common internal tangent of both circles. is The line of centers OO9 passes through P, is perpendicular to AB, and is equal in length to the sum of the radii, 4 R r. Also AB bisects each of the common external tangents, CD and EF. Circles Tangent Internally Circles O and O9 in Fig. 6-22 are tangent internally at P. AB is the common 4 external tangent of both circles. The line of centers OO9 if extended passes through P, is perpendicular to AB, and is equal in length to the difference of the radii, R r.
Fig. 6-22
Overlapping Circles Circles O and O9 in Fig.S 6-23 overlap. Their common chord is AB. If the circles are unequal, their (equal) comS mon external tangents CD and EF meet at P. The line of centers OO9 is the perpendicular bisector of AB and, if extended, passes through P. Circles Outside Each Other Circles O and O9 in Fig. 6-24 are entirely outside each other. The common internal tangents, AB and CD meet at P. If the circles are unequal, their common external tangents, EF and GH if extended, meet at P9. The line of centers OO9 passes through P and P9. Also, AB CD and EF GH.
Fig. 6-23
Fig. 6-24
CHAPTER 6 Circles
SOLVED PROBLEMS
6.5 Triangles and quadrilaterals having tangent sides Points P, Q, and R in Fig. 6-25 are points of tangency.
Fig. 6-25
(a) In Fig. 6-25(a), if AP (b) In Fig. 6-25(b), if AP (c) In Fig. 6-25(b), if AP
OP, what kind of triangle is OPA PQ, what kind of triangle is APQ OP, what kind of quadrilateral is OPAQ
(d) In Fig. 6-25(c), if OQ ' PR, what kind of quadrilateral is PABR
Solutions
(a) AP is tangent to the circle at P; then by Principle 1, j OPA is a right angle. Also, AP is an isosceles right triangle. (b) AP and AQ are tangents from a point to the circle; hence by Principle 4, AP ^ APQ is an equilateral triangle. (c) By Principle 4, AP Then AP AQ OP AQ. Also, OP and OQ are > radii. And AP OP. Hence, ^ OAP PQ. Then
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