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12-1 Tangent Lines

12-1 Tangent Lines. Understanding Tangent Lines. A tangent to a circle is a line that intersects a circle in exactly one point. The point where a circle and a tangent intersect is the point of tangency .

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12-1 Tangent Lines

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  1. 12-1 Tangent Lines

  2. Understanding Tangent Lines • A tangent to a circle is a line that intersects a circle in exactly one point. • The point where a circle and a tangent intersect is the point of tangency. Theorem 12-1: If a line is tangent to a circle, then the line is perpendicular to the radius at the point of tangency.

  3. Finding Angle Measures • ML and MN are tangent to O. What is the value of x? • ED is tangent to O. What is the value of x?

  4. Finding a Radius • What is the radius of C?  What is the radius of O?

  5. More Tangent Theorems Converse to Theorem 12-1: If a line is perpendicular to a radius at its endpoint on a circle, then the line is tangent to the circle. Theorem 12-3: If two tangent segments to a circle share a common endpoint outside the circle, then the two segments are congruent.

  6. Identifying a Tangent • Is ML tangent to N at L?  Use the diagram above. If NL = 4, ML = 7, and NM = 8, is ML a tangent?

  7. Circles Inscribed in Polygons • What is the perimeter of ΔABC?  If the perimeter of ΔPQR is 88 cm, what is QY?

  8. 12-2 Chords and Arcs

  9. Chords • A chord is a segment whose endpoints are on a circle. Theorem 12-4: Within a circle (or congruent circles), congruent central angles have congruent arcs. Converse: Within a circle (or congruent circles), congruent arcs have congruent central angles.

  10. More About Chords Theorem 12-5: Within a circle (or congruent circles), congruent central angles have congruent chords. Converse: With a circle (or congruent circles), congruent chords have congruent central angles. Theorem 12-6: Within a circle (or congruent circles), congruent chords have congruent arcs. Converse: With a circle (or congruent circles), congruent arcs have congruent chords.

  11. Using Congruent Chords In the diagram, O P. Given that BC DF, what can you conclude?

  12. Still More About Chords Theorem 12-7: Within a circle (or congruent circles), chords equidistant from the center(s) are congruent. Converse: Within a circle (or congruent circles), congruent chords are equidistant from the center(s).

  13. Finding the Length of a Chord • What is RS in O?  What is the value of x?

  14. Even More About Chords Theorem 12-8: In a circle, if a diameter is perpendicular to a chord, then it bisects the chord and its arc. Theorem 12-9: In a circle, if a diameter bisects a chord that is not the diameter, then it is perpendicular to the chord. Theorem 12-10: In a circle the perpendicular bisector of a chord contains the center of the circle.

  15. Finding Measures in a Circle • What is the value of r to the nearest tenth?

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