Warm-up 4.2 Identify each of the following from the diagram below. Center 3 radii 3 chords Secant

1. Warm-up 4.2 Identify each of the following from the diagram below. Center 3 radii 3 chords Secant Tangent Point of Tangency H B A G C J F E D

2. Warm-up 4.2 Identify each of the following from the diagram below. Center 3 radii 3 chords Secant Tangent Point of Tangency H B A G C J F E D

3. Properties of Tangents Section 4.2 Standard: MM2G3 ad Essential Question: How are tangents used to solve problems?

4. Recall: a tangent is a line in the plane of a circle that intersects the circle in exactly one point, the point of tangency. A tangent rayand a tangent segment are also called tangents.

5. Theorem 1: In a plane, a line is tangent to a circle if and only if the line is perpendicular to a radius of the circle at its endpoint on the circle (the point of tangency). For the figure at right, identify the center of the circle as O and the point of tangency as P. Mark a square corner to indicate that the tangent line is perpendicular to the radius. P O

6. Theorem 2 : Tangent segments from a common external point are congruent. Measure and with a straightedge to the nearest tenth of a cm. RS = _______ cm RT = ______ cm 2.6 2.6 S 2.6 cm R 2.6 cm T

7. Example 1: In the diagram below, is a radius of circle R. If TR = 26 , is tangent to circle R? Right Triangle? 102 + 242 = 262 676 = 676 Therefore, ∆RST is a right triangle. So, is tangent to . R 26 T 10 24 S

8. Example 2:is tangent to C at R and is tangent to C at S. Find the value of x. 32 = 3x + 5 27 = 3x 9 = x R 32 Q 3x + 5 S

9. Example 3: Find the value(s) of x: x2 x2 = 16 x = ±4 Q R 16 S

10. Example 4: In the diagram, B is a point of tangency. Find the length of the radius, r, of C. r2 + 702 = (r + 50)2 r2 + 4900 = r2 + 100r + 2500 2400 = 100r 24 = r C r 50 r 70 B

11. Recall: Two polygons are similar polygons if corresponding angles are congruent and corresponding sides are proportional. In the statement ABD DEF, the symbol  means “is similar to.”

12. Triangle Similarity Postulates and Theorems: Angle-Angle (AA) Similarity Postulate: If two angles of one triangle are congruent to two angles of another triangle, then the two triangles are similar. Side-Side-Side (SSS) Similarity Theorem: If the corresponding side lengths of two triangles are proportional, then the triangles are similar. Side-Angle-Side (SAS) Similarity Theorem: If an angle of one triangle is congruent to an angle of a second triangle and the lengths of the sides including these angles are proportional, then the triangles are similar.

13. Example 5: In the diagram, the circles are concentric with center A. is tangent to the inner circle at B and is tangent to the outer circle at C. Use similar triangles to show that . A E B D C

14. tangent iff to radius 1. 1. ________________   2. _____________________ 2. Definition of   3. _____________________ 3. All right angles are   4. CAD BAE 4. _________________  5. _____________________ 5. AA Similarity Postulate 6. _____________________ 6. Corresponding lengths of similar triangles are in proportion A E B D C

15. Example 6: In the diagram, is a common internal tangent to M and P. Use similar triangles to show that P S N T M

16. tangent iff to radius P 1. ________________ 2. _____________________ 2. Definition of  3. _____________________ 3. All right angles are   4. MNS PNT 4. ________________  5. _____________________ 5. AA Similarity Postulate 6. _____________________ 6. Corresponding lengths of similar triangles are in proportion S N T M

17. Example 7: Use the diagram at right to find each of the following: 1. Find the length of the radius of A. 2. Find the slope of the tangent line, t. t A (3, 1) (5, -1)