1. t Circles – Tangent Lines
A tangent line touches a circle at exactly one point.
In this case, line t is tangent to circle A. A t

2. t Circles – Tangent Lines
A tangent line touches a circle at exactly one point.
In this case, line t is tangent to circle A.
The point at which the line is tangent is called the point of tangency ( point C ) A D C t B

3. t Circles – Tangent Lines
A tangent line touches a circle at exactly one point.
In this case, line t is tangent to circle A.
The point at which the line is tangent is called the point of tangency ( point C )
Rays can also be tangent to circles.
Ray CD ( segment CD )
Ray CB ( segment CB ) A D C t B

4. t a Circles – Tangent Lines
A tangent line touches a circle at exactly one point.
In this case, line t is tangent to circle A.
The point at which the line is tangent is called the point of tangency ( point C )
Rays can also be tangent to circles.
Ray CD ( segment CD )
Ray CB ( segment CB )
Common Tangent Line
- tangent to two coplanar circles A D C t B a P Q

5. t a e Circles – Tangent Lines
A tangent line touches a circle at exactly one point.
In this case, line t is tangent to circle A.
The point at which the line is tangent is called the point of tangency ( point C )
Rays can also be tangent to circles.
Ray CD ( segment CD )
Ray CB ( segment CB )
Common Tangent Line
- tangent to two coplanar circles
- common external tangent lines do
not intersect ( lines “a” and “e” ) the
segment joining the circles A D C t B a P Q e

6. t g h a e Circles – Tangent Lines
A tangent line touches a circle at exactly one point.
In this case, line t is tangent to circle A.
The point at which the line is tangent is called the point of tangency ( point C )
Rays can also be tangent to circles.
Ray CD ( segment CD )
Ray CB ( segment CB )
Common Tangent Line
- tangent to two coplanar circles
- common external tangent lines do
not intersect ( lines “a” and “e” ) the
segment joining the circles
common internal tangent lines
intersect the segment joining the circles A D C t B g h a P Q e

7. c t Circles – Tangent Lines Tangent circles
- two coplanar circles that are tangent to the same line at the same point c A C Q S B t

8. c t Circles – Tangent Lines Tangent circles
- two coplanar circles that are tangent to the same line at the same point
circle A tangent to circle Q
circle A tangent to circle S c A C Q S B t
Circle A and Q are internally tangent, one circle is inside the other.

9. c t Circles – Tangent Lines Tangent circles
- two coplanar circles that are tangent to the same line at the same point
circle A tangent to circle Q
circle A tangent to circle S c A C Q S B t
Circle A and Q are internally tangent, one circle is inside the other.
Circle A and S are externally tangent, not one point of one circle is in the interior of the other.

10. t Circles – Tangent Lines
Theorem – the radius of a circle is perpendicular to a tangent line of that circle drawn to the point of tangency. A B t

11. t Circles – Tangent Lines
Theorem – the radius of a circle is perpendicular to a tangent line of that circle drawn to the point of tangency. A B t
Theorem - tangents to a circle from an exterior point are congruent D A P E

12. t Circles – Tangent Lines
Theorem – the radius of a circle is perpendicular to a tangent line of that circle drawn to the point of tangency. A B t
Theorem - tangents to a circle from an exterior point are congruent D A P E

13. Circles – Tangent Lines
Theorem – the radius of a circle is perpendicular to a tangent line of that circle drawn to the point of tangency.
Theorem - tangents to a circle from an exterior point are congruent
Let’s use these two theorems to solve some problems.

14. Circles – Tangent Lines
Theorem – the radius of a circle is perpendicular to a tangent line of that circle drawn to the point of tangency.
Theorem - tangents to a circle from an exterior point are congruent
Let’s use these two theorems to solve some problems.
AB and AC are tangent to circle S. B A S C

15. Circles – Tangent Lines
Theorem – the radius of a circle is perpendicular to a tangent line of that circle drawn to the point of tangency.
Theorem - tangents to a circle from an exterior point are congruent
Let’s use these two theorems to solve some problems.
AB and AC are tangent to circle S. B A S C

16. Circles – Tangent Lines
Theorem – the radius of a circle is perpendicular to a tangent line of that circle drawn to the point of tangency.
Theorem - tangents to a circle from an exterior point are congruent
Let’s use these two theorems to solve some problems.
AB and AC are tangent to circle S. B A S C
( 90° - 25° = 65° )

17. Circles – Tangent Lines
Theorem – the radius of a circle is perpendicular to a tangent line of that circle drawn to the point of tangency.
Theorem - tangents to a circle from an exterior point are congruent
Let’s use these two theorems to solve some problems.
AB and AC are tangent to circle S.
∆ABC is isosceles from the theorem above about tangents from an exterior point… B A S C
( 90° - 25° = 65° )

18. Circles – Tangent Lines
Theorem – the radius of a circle is perpendicular to a tangent line of that circle drawn to the point of tangency.
Theorem - tangents to a circle from an exterior point are congruent
Let’s use these two theorems to solve some problems.
AB and AC are tangent to circle S.
∆ABC is isosceles from the theorem above about tangents from an exterior point… B
65° A S
65° C
( 90° - 25° = 65° )

19. Circles – Tangent Lines
Theorem – the radius of a circle is perpendicular to a tangent line of that circle drawn to the point of tangency.
Theorem - tangents to a circle from an exterior point are congruent
Let’s use these two theorems to solve some problems.
AB and AC are tangent to circle S.
∆ABC is isosceles from the theorem above about tangents from an exterior point… B
65° A S
65° C
( 90° - 25° = 65° )

20. SC BA AC Circles – Tangent Lines
Theorem – the radius of a circle is perpendicular to a tangent line of that circle drawn to the point of tangency.
Theorem - tangents to a circle from an exterior point are congruent
EXAMPLE # 2 :
AB and AC are tangent to circle S.
FIND : B SC BA AC 10 A S 26 C

21. SC 10 BA AC Circles – Tangent Lines
Theorem – the radius of a circle is perpendicular to a tangent line of that circle drawn to the point of tangency.
Theorem - tangents to a circle from an exterior point are congruent
EXAMPLE # 2 :
AB and AC are tangent to circle S.
FIND : B SC 10 BA AC 10 A S 26 C

22. SC 10 BA 24 AC Circles – Tangent Lines
Theorem – the radius of a circle is perpendicular to a tangent line of that circle drawn to the point of tangency.
Theorem - tangents to a circle from an exterior point are congruent
EXAMPLE # 2 :
AB and AC are tangent to circle S.
FIND : B SC 10 BA 24 AC 10 A S 26
∆BAS is a right triangle : C

23. SC 10 BA 24 AC Circles – Tangent Lines
Theorem – the radius of a circle is perpendicular to a tangent line of that circle drawn to the point of tangency.
Theorem - tangents to a circle from an exterior point are congruent
EXAMPLE # 2 :
AB and AC are tangent to circle S.
FIND : B SC 10 BA 24 AC 10 A S 26
∆BAS is a right triangle : C

24. Circles – Tangent Lines
Theorem – the radius of a circle is perpendicular to a tangent line of that circle drawn to the point of tangency. C
EXAMPLE # 3 :
10.5
AD is tangent to circle S.
Find CD if CS = 10.5 and AD = 25 S A 25 D

25. Circles – Tangent Lines
Theorem – the radius of a circle is perpendicular to a tangent line of that circle drawn to the point of tangency. C
EXAMPLE # 3 :
10.5
AD is tangent to circle S.
Find CD if CS = 10.5 and AD = 25 S
10.5
( both are radii ) A 25 D

26. Circles – Tangent Lines
Theorem – the radius of a circle is perpendicular to a tangent line of that circle drawn to the point of tangency. C
EXAMPLE # 3 :
10.5
AD is tangent to circle S.
Find CD if CS = 10.5 and AD = 25 S
10.5
( both are radii ) A 25 D

27. Circles – Tangent Lines
Theorem – the radius of a circle is perpendicular to a tangent line of that circle drawn to the point of tangency. C
EXAMPLE # 3 :
10.5
AD is tangent to circle S.
Find CD if CS = 10.5 and AD = 25 S
10.5
( both are radii ) A 25 D

28. Circles – Tangent Lines
Theorem – the radius of a circle is perpendicular to a tangent line of that circle drawn to the point of tangency. C D
EXAMPLE # 4 : 14 24
CD is a common tangent to circle S and circle R.
Find CD if CS = 24, DR = 14
and SR = 26 R 26 S

29. Circles – Tangent Lines
Theorem – the radius of a circle is perpendicular to a tangent line of that circle drawn to the point of tangency. C D
EXAMPLE # 4 : 14 24 E
CD is a common tangent to circle S and circle R.
Find CD if CS = 24, DR = 14
and SR = 26 R 26 S
We can sketch in a parallel line to CD that creates a right triangle SRE.
Since CD was perpendicular to CS and DR, the new line will be perpendicular as well.

30. Circles – Tangent Lines
Theorem – the radius of a circle is perpendicular to a tangent line of that circle drawn to the point of tangency. C D
EXAMPLE # 4 : 14 24 E
CD is a common tangent to circle S and circle R.
Find CD if CS = 24, DR = 14
and SR = 26 10 R 26 S
We can sketch in a parallel line to CD that creates a right triangle SRE.
Since CD was perpendicular to CS and DR, the new line will be perpendicular as well.

31. Circles – Tangent Lines
Theorem – the radius of a circle is perpendicular to a tangent line of that circle drawn to the point of tangency. C D
EXAMPLE # 4 : 14 24 E
CD is a common tangent to circle S and circle R.
Find CD if CS = 24, DR = 14
and SR = 26 10 R 26 S

32. Circles – Tangent Lines
Theorem – the radius of a circle is perpendicular to a tangent line of that circle drawn to the point of tangency. C D
EXAMPLE # 4 : 14 24 E
CD is a common tangent to circle S and circle R.
Find CD if CS = 24, DR = 14
and SR = 26 10 R 26 S