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Section 6.3: Line and Segment Relationships in the Circle.

Learn the key theorems and concepts related to lines and segments in circles, including tangent circles, common tangents, and lengths of segments. Practice proofs and examples to deepen your understanding.

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Section 6.3: Line and Segment Relationships in the Circle.

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  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 6.3.3 The perpendicular bisector of a chord contains the center of the circle. Ex. 1 p. 301 Section 6.3 Nack

  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) Section 6.3 Nack

  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. Section 6.3 Nack

  4. Tangents External to a Circle • Theorem 6.3.4: The tangent segments to a circle from an external point are congruent. Fig. 6.46 p. 303 Ex 2,3 p. 303 Section 6.3 Nack

  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 Ex. 4,5 p. 304-5 Section 6.3 Nack

  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. Ex. 6 p. 306 Section 6.3 Nack

  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. Ex. 7 p. 306 Section 6.3 Nack

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