Section 12.1: Lines That intersect Circles

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

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

3. 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 Words and Phrases to Remember

4. Example of the Lines and Segments

5. Concentric Circles Tangent Circles Examples of Pairs of Circles

6. Common Tangent

7. 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. How to Identify Tangents of Circles

8. 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. THEOREMS

9. Pythagorean Thm. c2 = a2 + b2 (r + 8)2 = r2 + 162 Substitute values Square of binomial r 2 + 16r + 64 = r2 + 256 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. 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