12-1 Tangent Lines

Quick Review

A tangent to a circle is a line that intersects the circle at exactly one point. The radius to that point is perpendicular to the tangent. From any point outside a circle, you can draw two segments tangent to a circle. Those segments are congruent.

Example

P A → p eh vector  and P B → p b vector  are tangents. Find x.

The radii are perpendicular to the tangents. Add the angle measures of the quadrilateral:

x + 90 + 90 + 40 = 360 x + 220 = 360 x = 140 table with 3 rows and 1 column , row1 column 1 , x plus 90 plus 90 plus 40 equals 360 , row2 column 1 , x plus 220 equals 360 , row3 column 1 , x equals 140 , end table

Exercises

Use ⊙ O circle dot o  for Exercises 6–8.

6. What is the perimeter of Δ A B C ? cap delta eh b c , question mark
7. O B = 28 . o b equals square root of 28 .  What is the radius?
8. What is the value of x?

12-2 Chords and Arcs

Quick Review

A chord is a segment whose endpoints are on a circle. Congruent chords are equidistant from the center. A diameter that bisects a chord that is not a diameter is perpendicular to the chord. The perpendicular bisector of a chord contains the center of the circle.

Example

What is the value of d?

Since the chord is bisected, m ∠ A C B = 90 . m angle . eh c b equals 90 .  The radius is 13 units. So an auxiliary segment from A to B is 13 units. Use the Pythagorean Theorem.

d 2 + 12 2 = 13 2 d 2 = 25 d = 5 table with 3 rows and 2 columns , row1 column 1 , d squared , plus , 12 squared , column 2 equals , 13 squared , row2 column 1 , d squared , column 2 equals 25 , row3 column 1 , d , column 2 equals 5 , end table

Exercises

Use the figure below for Exercises 9–11.

9. If A B ¯ eh b bar  is a diameter and C E = E D , c e equals e d comma  then m ∠ A E C = ? ¯ m angle . eh e c equals , modified question mark with under bar below
10. If A B ¯ eh b bar  is a diameter and is at right angles to C D ¯ , c d bar , comma  what is the ratio of CD to DE?

11. If C E = 1 2 C D c e equals . 1 half c d  and m ∠ D E B = 90 , m angle . d e b equals 90 comma  what is true of A B ¯ ? eh b bar , question mark

    

Use the circle below for Exercises 12 and 13.

12. What is the value of x?

13. What is the value of y?

    

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