1. Section 12.1: Lines That intersect Circles
By: The Balloonicorns

2. Stuff to learn; Identify tangents, secants, and chords
Use properties of tangents to solve problems

3. Words and Phrases to Remember
Interior of a Circle – The set of all points inside the circle
Exterior of a Circle – The set of all points outside the circle
Chord – A segment whose endpoints lie on a circle
Secant – A line that intersects a circle at two points
Tangent of a Circle – A line in the same plane as a circle that intersects it at exactly one point
Point of Tangency – The point where the tangent and circle intersect
Congruent Circles – Two circles that have congruent radii
Concentric Circles – Coplanar circles with the same center
Tangent Circles – Two coplanar circles that intersect at exactly one point
Common Tangent – A line that is tangent to two circles

4. Example of the Lines and Segments

5. Examples of Pairs of Circles
Concentric Circles
Tangent Circles

6. Common Tangent

7. How to Identify Tangents of Circles
Center of circle A is (4, 4), and its radius is 4. The center of circle B is (5, 4) and its radius is 3. The two circles have one point of intersection (8, 4). The vertical line x = 8 is the only common tangent of the two circles.

8. THEOREMS
12-1-1: If a line is tangent to a circle, then it is perpendicular to the radius drawn to the point of tangency.
12-1-2: If a line is perpendicular to a radius of a circle at a point on the circle, then the line is tangent to the circle.
12-1-3: If two segments are tangent to a circle from the same external point, then the segments are congruent.

9. How To use Tangents c2 = a2 + b2 Pythagorean Thm. (r + 8)2 = r2 + 162
Substitute values
r r + 64 = r
Square of binomial
16r + 64 = 256
Subtract r2 from each side.
16r = 192
Subtract 64 from each side
r = 12
Divide.
How To use Tangents

10. Practice

11. Using Properties of Tangents
AB = AD
Two tangent segments from the same point are 
11 = x2 + 2
Substitute values
9 = x2
Subtract 2 from each side.
3 = x
Find the square root of 9.
Using Properties of Tangents

12. Practice