Warm-up 4.2 Identify each of the following from the diagram below. 1.Center 2.3 radii 3.3 chords 4.Secant 5.Tangent 6.Point of Tangency C A B D E G H F J

Warm-up 4.2 Identify each of the following from the diagram below. 1.Center 2.3 radii 3.3 chords 4.Secant 5.Tangent 6.Point of Tangency C A B D E G H F J

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

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 ray and a tangent segment are also called tangents.

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. O P

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 T S R 2.6 cm 2.6

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

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

Example 3: Find the value(s) of x : S R Q x2x2 16 x 2 = 16 x = ±4

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

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.”

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.

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 B C D E

1. 1. ________________ 2. _____________________2. Definition of  3. _____________________3. All right angles are  4.  CAD   BAE4. _________________ 5. _____________________5. AA Similarity Postulate 6. _____________________6. Corresponding lengths of similar triangles are in proportion tangent iff  to radius A B C D E

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

1.1. ________________ 2. _____________________2. Definition of  3. _____________________3. All right angles are  4.  MNS   PNT4. ________________ 5. _____________________5. AA Similarity Postulate 6. _____________________6. Corresponding lengths of similar triangles are in proportion M N T P S tangent iff  to radius

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. A (3, 1) (5, -1) t