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?