Use Properties of Tangents 6.1 Use Properties of Tangents Example 1 Identify special segments and lines F B E A C D Tell whether the line, ray, or segment is best described as a radius, chord, diameter, secant or tangent of C? Solution BC is a _________ because C is the center and B is a point on the circle. radius EA is a _________ because it is a line that intersects the circle in two points. secant DE is a _________ ray because it is contained in a line that intersects the circle in exactly one point. tangent

Use Properties of Tangents 6.1 Use Properties of Tangents Example 2 Find lengths in circles in a coordinate plane Use the diagram to find the given lengths. B Radius of A A Diameter of A Radius of B Diameter of B Solution The radius of A is ___ units. 2 The diameter of A is ___ units. 4 The radius of B is ___ units. 4 The diameter of B is ___ units. 8

Use Properties of Tangents 6.1 Use Properties of Tangents Checkpoint. Complete the following exercises. F B E A C D In Example 1, tell whether AB is best described as a radius, chord, diameter, secant, or tangent. Explain. AB is a diameter because it is a chord that contains the center C.

Use Properties of Tangents 6.1 Use Properties of Tangents Checkpoint. Complete the following exercises. Use the diagram to find (a) the radius of C and (b) the diameter of D. D C The radius of C is 3 units. The diameter of D is 2 units.

Use Properties of Tangents 6.1 Use Properties of Tangents Example 3 Draw common tangents Tell how many common tangents the circles have and draw them. a. b. c. Solution ___ common tangents 3 ___ common tangents 2 ___ common tangents 1

Use Properties of Tangents 6.1 Use Properties of Tangents Checkpoint. Tell how many common tangents the circles have and draw them. 3. 4. no common tangents 4 common tangents

Use Properties of Tangents 6.1 Use Properties of Tangents Theorem 6.1 If a plane, a line is tangent to a circle if and only if the line is _____________ to the radius of the circle at its endpoint on the circle. perpendicular m O P

Use Properties of Tangents 6.1 Use Properties of Tangents T R S Example 4 Verify a tangent to a circle In the diagram, RS is a radius of R. Is ST tangent to R? Solution Use the Converse of the Pythagorean Theorem. Because 102 + 242 = 262, RST is a _____________ and RS ____. right triangle So, _____ is perpendicular to a radius of R at its endpoint on R. By ____________, ST is _________ to R. Theorem 6.1 tangent

Use Properties of Tangents 6.1 Use Properties of Tangents Checkpoint. RS is a radius of R. Is ST tangent to R? 5. R S 5 T 12 8 13 Therefore, RS ST. By Theorem 6.1, ST is tangent to R.

Use Properties of Tangents 6.1 Use Properties of Tangents Checkpoint. RS is a radius of R. Is ST tangent to R? T S R 12 16 7 6. 19 NO

Use Properties of Tangents 6.1 Use Properties of Tangents B C A Example 5 Find the radius of a circle In the diagram, B is a point of tangency. Find the radius r of C. Solution You know from Theorem 6.1 that AB BC, so ABC is a _____________. You can use Pythagorean Theorem. right triangle Pythagorean Theorem Substitute. Multiply. Subtract from each side. 98 Divide by ____. The radius of C is _____. 36

Use Properties of Tangents 6.1 Use Properties of Tangents Checkpoint. Complete the following exercises. J L K In the diagram, K is a point of tangency. Find the radius r of L.

Use Properties of Tangents 6.1 Use Properties of Tangents Theorem 6.2 Tangent segments from a common external point are _____________. congruent R P S T

Use Properties of Tangents 6.1 Use Properties of Tangents Example 6 Use properties of tangents C Q R S QR is tangent to C at R and QS is tangent to C at S. Find the value of x. Solution Tangent segments from a common external point are ___________. congruent Substitute. Solve for x.

Use Properties of Tangents 6.1 Use Properties of Tangents Triangle Similarity Postulates and Theorems Angle-Angle (AA) Similarity Postulate: If two angles of one triangle are ___________ to two angles of another _________, then the two triangles are _________. congruent triangle similar Theorem 6.3 Side-Side-Side (SSS) Similarity Theorem: If the corresponding side lengths of two triangles are _____________, then the triangles are _________. proportional similar Theorem 6.4 Side-Angle-Side (SAS) Similarity Theorem: If an angle of one triangle is _____________ to an angle of a second triangle and the lengths of the sides including these angles are ______________, then the triangles are ________. congruent proportional similar

Use Properties of Tangents 6.1 Use Properties of Tangents Example 7 Use tangents with similar triangles In the diagram, both circles are centered at A. BE is tangent to the inner circle at B and CD is tangent to the outer circle at C. Use similar triangles to show that A B C E D Solution ______________ Theorem 6.1 Definition of . All right are . Reflexive Prop ______________ AA Similarity Post Corr. sides lengths are prop.

Use Properties of Tangents 6.1 Use Properties of Tangents Checkpoint. Complete the following exercises. C T R S RS is tangent to C at S and RT is tangent to C at T. Find the value(s) of x.

Use Properties of Tangents 6.1 Use Properties of Tangents Pg. 198, 6.1 #1-34