1. CIRCLES
Chapter 10

2. Tangents to Circles lesson 10.1
California State Standards
7: Prove and Use theorems involving
properties of circles.
21: Prove and Solve relationships among chords,
secants and tangents.

3. definitions Circle Center Radius
The set of all points in a plane that are equidistant
from a given point.
Center
The given point.
Radius
A line segment with the center as one endpoint and
a point on the circle as the other endpoint.
The distance from the center to a point on the circle.

4. the plural of radius is radii
definitions P
Circle
The set of all points in a plane that are equidistant
from a given point.
Center
The given point.
Radius
A line segment with the center as one endpoint and
a point on the circle as the other endpoint.
The distance from the center to a point on the circle. C
the plural of radius is radii

5. definitions Congruent Circles Diameter Circles with the same radius
A line segment with endpoints on the circle
that contains the center of the circle.
The distance across a circle through the center.
2r = d

6. 2r = d definitions Congruent Circles Diameter
Circles with the same radius
Diameter
A line segment with endpoints on the circle
that contains the center of the circle.
The distance across a circle through the center.
2r = d P C R
2r = d

7. definitions Chord Secant Tangent
A segment whose endpoints are on the circle.
A diameter is a “specialized” chord.
Secant
A line that intersects a circle in two points.
Tangent
A line that intersects a circle in exactly one point.
The circle and line must lie in the same plane.

8. definitions T U Chord B Secant Tangent C A F G
A segment whose endpoints are on the circle.
A diameter is a “specialized” chord.
Secant
A line that intersects a circle in two points.
Tangent
A line that intersects a circle in exactly one point.
The circle and line must lie in the same plane. B C A F G

9. Identify each line or segment
secant A
chord Q
radius
tangent
diameter
radius P C X
radius
chord B
Identify each line or segment

10. definition Tangent Circles externally tangent circles
Coplanar circles that intersect in exactly one point.
externally tangent circles

11. definition Tangent Circles internally tangent circles
Coplanar circles that intersect in exactly one point.
internally tangent circles

12. definition
Concentric Circles
Coplanar circles with a common center.

13. definition Common Tangent A line or segment that is tangent to two
coplanar circles
Common Internal Tangent crosses between the circles
Common External Tangent stays along the edges of the circles

14. definition Common Internal Tangent Common Tangent
A line or segment that is tangent to two
coplanar circles
Common Internal Tangent crosses between the circles
Common External Tangent stays along the edges of the circles
Common External Tangent

15. example
Is the common tangent
internal or external? D C T
external

16. example tangent diameter chord radius secant Describe each segment H EF
chord I A
radius
secant D

17. definitions Interior of a Circle Exterior of a Circle
The set of points inside the circle
Exterior of a Circle
The set of points outside the circle.
interior
exterior

18. theorem t Circle Tangent-Radius Perpendicular
If a line is tangent to a circle,
then it is perpendicular to the radius
drawn to the point of tangency. P t C

19. theorem t Circle Tangent-Radius Perpendicular Converse
If a line is perpendicular to a radius of
a circle at the endpoint on the circle,
then the line is tangent to the circle. P t C

20. example B 11 43 C 45 A

21. example C r r 16 A 24 B

22. theorem Congruent Tangents If two segments from the same exterior
point are tangent to a circle,
then the segments are congruent. P S C Q

23. example D
x2 – 4 C A 21 B

24. example D 15 C x 36 z y 15 A 36 B

25. example D 15 C 36 39 z 15 A 36 B

26. example What are the coordinates of each center? y xB x
What is the radius of each circle?

27. example
Describe the intersection of the two circles. y A B x

28. example
Describe the common tangents of the circles. y A B x

29. example What are the coordinates of each center? y
What is the radius of each circle? A B
Describe any common tangents. x

30. C A B D
Statement
Reason