12-1 Tangent Lines.

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Presentation transcript:

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. Theorem 12-1: If a line is tangent to a circle, then the line is perpendicular to the radius at the point of tangency.

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?

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

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.

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?

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

12-2 Chords and Arcs

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.

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.

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

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).

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

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.

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