1. Section 6.3: Line and Segment Relationships in the Circle.
Theorem 6.3.1: If a line is drawn through the center of a circle perpendicular to a chord, then it bisects the chord and its arc.
Theorem 6.3.2: If a line through the center of a circle bisects a chord other than a diameter, then it is perpendicular to the chord.
Theorem The perpendicular bisector of a chord contains the center of the circle.
Ex. 1 p. 297 O
R M S
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2. Circles that are Tangent
Two circles that touch at one point.
P and Q are internally tangent.
O and R are externally tangent.
Line of centers is the line or line segment containing the centers of two circles with different centers. OR in (b)
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3. Common Tangent Lines to Circles
Common Tangent is a line segment that is tangent to each of two circles.
Common External Tangent does not intersect the line of centers.
Common Internal Tangent does intersect the line of centers.
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4. Tangents External to a Circle
Theorem 6.3.4: The tangent segments to a circle
from an external point are congruent. Fig p. 299
Ex 2,3 p. 299
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5. Lengths of Segments in a Circle
Theorem 6.3.5: If two chords intersect within a circle then the product of the lengths of the segments of one chord is equal to the product of the lengths of the segments of the other chord.
Prove: RV  VS = TVVQ
Draw RT and QS.
1  2 by Vertical Angles
R and Q inscribed angles intersecting the same are so R  Q
By AA, ΔRTV ~ ΔQSV
RV = TV by CSSTP so RV  VS = TVVQ
VQ VS
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6. Tangents and Secants External to the Circle
Theorem 6.3.6: If two secant segments are drawn to a circle from an external point, then the products of the lengths of each secant with its external segment are equal.
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7. Theorem 6.3.7: If a tangent segment and a secant segment are drawn to a circle from an external point, then the square of the length of the tangent equals the product of the length of the secant, with the length of its external segment.
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