Essential Questions

Friday, January 22 Essential Questions • How do I use the properties of tangents to identify lengths in a circle? • How do you use information of a circle to find arc measures?

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

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

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

6.1 Use Properties of Tangents • 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. Checkpoint. Complete the following exercises.

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

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

6.1 Use Properties of Tangents O m 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 P

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

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

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

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

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

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

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

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

6.1 Use Properties of Tangents 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 E B D C Definition of . All right are . Use tangents with similar triangles Example 7 Solution Theorem 6.1 ______________ Reflexive Prop ______________ AA Similarity Post Corr. sides lengths are prop.

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

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

Essential Questions