1. 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?

2. 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

3. 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

4. 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.

5. 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.

6. 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

7. 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

8. 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

9. 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 = 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

10. 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.

11. 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

12. 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

13. 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.

14. 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

15. 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.

16. 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

17. 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.

18. 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.

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