Tangent Lines Understanding Tangent Lines A tangent to

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Tangent Lines

Tangent Lines

Understanding Tangent Lines • A tangent to a circle is a line that intersects

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

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

Finding a Radius • What is the radius of C? What is the radius of O?

More Tangent Theorems Theorem 12 -2: If two tangent segments to a circle share

More Tangent Theorems Theorem 12 -2: 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

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

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

Chords and Arcs

Chords and Arcs

Chords • A chord is a segment whose endpoints are on a circle. Theorem

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

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

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

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).

Even More About Chords Theorem 12 -8: In a circle, if a diameter is

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 the Length of a Chord • What is RS in O? What is

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

Finding Measures in a Circle • What is the value of r to the

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

Solve for x, QT UV = 6; RS = 3; ST = 3

Solve for x, QT UV = 6; RS = 3; ST = 3

Solve for x, QT UV = 6; RS = 3; ST = 3 x

Solve for x, QT UV = 6; RS = 3; ST = 3 x = 4, Since Congruent chord are an equal distance from the center.

Solve for x, QT UV = 6; RS = 3; ST = 3 x

Solve for x, QT UV = 6; RS = 3; ST = 3 x = 4,

Find the measure of the arc Solve for x and y

Find the measure of the arc Solve for x and y

Find the measure of the arc Solve for x and y

Find the measure of the arc Solve for x and y

D 2 x° (x + 40)°

D 2 x° (x + 40)°

Ex. 3: Using Theorem 10. 4 AB = 8; DE = 8, and CD

Ex. 3: Using Theorem 10. 4 AB = 8; DE = 8, and CD = 5. Find CF.

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