Tangents

1. Section 10-5
Tangents
Monday, May 21, 2012

2. Essential Questions
• How do you use properties of tangents?
• How do you solve problems involving
circumscribed polygons?
Monday, May 21, 2012

3. Vocabulary
1. Tangent:
2. Point of Tangency:
3. Common Tangent:
Monday, May 21, 2012

4. Vocabulary
1. Tangent: A line in the same plane as a circle and
intersects the circle in exactly one point
2. Point of Tangency:
3. Common Tangent:
Monday, May 21, 2012

5. Vocabulary
1. Tangent: A line in the same plane as a circle and
intersects the circle in exactly one point
2. Point of Tangency: The point that a tangent
intersects a circle
3. Common Tangent:
Monday, May 21, 2012

6. Vocabulary
1. Tangent: A line in the same plane as a circle and
intersects the circle in exactly one point
2. Point of Tangency: The point that a tangent
intersects a circle
3. Common Tangent: A line, ray, or segment that is
tangent to two different circles in the same plane
Monday, May 21, 2012

7. Theorems
Theorem 10.10 - Tangents:
Theorem 11.11 - Two Segments:
Monday, May 21, 2012

8. Theorems
Theorem 10.10 - Tangents: A line is tangent to a
circle IFF it is perpendicular to a radius drawn to
the point of tangency
Theorem 11.11 - Two Segments:
Monday, May 21, 2012

9. Theorems
Theorem 10.10 - Tangents: A line is tangent to a
circle IFF it is perpendicular to a radius drawn to
the point of tangency
Theorem 11.11 - Two Segments: If two segments
from the same exterior point are tangent to a
circle, then the segments are congruent
Monday, May 21, 2012

10. Example 1
Draw in the common tangents for each figure. If no tangent
exists, state no common tangent.
a. b.
Monday, May 21, 2012

11. Example 1
Draw in the common tangents for each figure. If no tangent
exists, state no common tangent.
a. b.
No common tangent
Monday, May 21, 2012

12. Example 1
Draw in the common tangents for each figure. If no tangent
exists, state no common tangent.
a. b.
No common tangent
Monday, May 21, 2012

13. Example 1
Draw in the common tangents for each figure. If no tangent
exists, state no common tangent.
a. b.
No common tangent
Monday, May 21, 2012

14. Example 2
KL is a radius of ⊙K. Determine whether LM is tangent to
⊙K. Justify your answer.
Monday, May 21, 2012

15. Example 2
KL is a radius of ⊙K. Determine whether LM is tangent to
⊙K. Justify your answer.
a +b =c
2 2 2
Monday, May 21, 2012

16. Example 2
KL is a radius of ⊙K. Determine whether LM is tangent to
⊙K. Justify your answer.
a +b =c
2 2 2
20 + 21 = 29
2 2 2
Monday, May 21, 2012

17. Example 2
KL is a radius of ⊙K. Determine whether LM is tangent to
⊙K. Justify your answer.
a +b =c
2 2 2
20 + 21 = 29
2 2 2
400 + 441 = 841
Monday, May 21, 2012

18. Example 2
KL is a radius of ⊙K. Determine whether LM is tangent to
⊙K. Justify your answer.
a +b =c
2 2 2
20 + 21 = 29
2 2 2
400 + 441 = 841
Since the values hold for a right triangle,
we know that LM is perpendicular to
KL, making it a tangent.
Monday, May 21, 2012

19. Example 3
In the figure, WE is tangent to ⊙D at W. Find the value of x.
Monday, May 21, 2012

20. Example 3
In the figure, WE is tangent to ⊙D at W. Find the value of x.
a +b =c
2 2 2
Monday, May 21, 2012

21. Example 3
In the figure, WE is tangent to ⊙D at W. Find the value of x.
a +b =c
2 2 2
x + 24 = ( x + 16)
2 2 2
Monday, May 21, 2012

22. Example 3
In the figure, WE is tangent to ⊙D at W. Find the value of x.
a +b =c
2 2 2
x + 24 = ( x + 16)
2 2 2
x + 576 = x + 32x + 256
2 2
Monday, May 21, 2012

23. Example 3
In the figure, WE is tangent to ⊙D at W. Find the value of x.
a +b =c
2 2 2
x + 24 = ( x + 16)
2 2 2
x2 + 576 = x 2 + 32x + 256
2 2
−x −x
Monday, May 21, 2012

24. Example 3
In the figure, WE is tangent to ⊙D at W. Find the value of x.
a +b =c
2 2 2
x + 24 = ( x + 16)
2 2 2
x2 + 576 = x 2 + 32x + 256
2 2
−x −256 −x −256
Monday, May 21, 2012

25. Example 3
In the figure, WE is tangent to ⊙D at W. Find the value of x.
a +b =c
2 2 2
x + 24 = ( x + 16)
2 2 2
x2 + 576 = x 2 + 32x + 256
2 2
−x −256 −x −256
320 = 32x
Monday, May 21, 2012

26. Example 3
In the figure, WE is tangent to ⊙D at W. Find the value of x.
a +b =c
2 2 2
x + 24 = ( x + 16)
2 2 2
x2 + 576 = x 2 + 32x + 256
2 2
−x −256 −x −256
320 = 32x
x = 10
Monday, May 21, 2012

27. Example 4
AC and BC are tangent to ⊙Z. Find the value of x.
Monday, May 21, 2012

28. Example 4
AC and BC are tangent to ⊙Z. Find the value of x.
3x + 2 = 4x − 3
Monday, May 21, 2012

29. Example 4
AC and BC are tangent to ⊙Z. Find the value of x.
3x + 2 = 4x − 3
x=5
Monday, May 21, 2012

30. Example 5
Some round cookies are marketed in a triangular package
to pique the consumer’s interest. If ∆QRS is
circumscribed with ⊙T with NQ = 2 cm, QR = 8 cm, and
SM = 10 cm, find the perimeter of ∆QRS.
Monday, May 21, 2012

31. Example 5
Some round cookies are marketed in a triangular package
to pique the consumer’s interest. If ∆QRS is
circumscribed with ⊙T with NQ = 2 cm, QR = 8 cm, and
SM = 10 cm, find the perimeter of ∆QRS.
2
8
10
Monday, May 21, 2012

32. Example 5
Some round cookies are marketed in a triangular package
to pique the consumer’s interest. If ∆QRS is
circumscribed with ⊙T with NQ = 2 cm, QR = 8 cm, and
SM = 10 cm, find the perimeter of ∆QRS.
2
10
8
10
Monday, May 21, 2012

33. Example 5
Some round cookies are marketed in a triangular package
to pique the consumer’s interest. If ∆QRS is
circumscribed with ⊙T with NQ = 2 cm, QR = 8 cm, and
SM = 10 cm, find the perimeter of ∆QRS.
2
10
2
8
10
Monday, May 21, 2012

34. Example 5
Some round cookies are marketed in a triangular package
to pique the consumer’s interest. If ∆QRS is
circumscribed with ⊙T with NQ = 2 cm, QR = 8 cm, and
SM = 10 cm, find the perimeter of ∆QRS.
2
10
2
8
10
6
Monday, May 21, 2012

35. Example 5
Some round cookies are marketed in a triangular package
to pique the consumer’s interest. If ∆QRS is
circumscribed with ⊙T with NQ = 2 cm, QR = 8 cm, and
SM = 10 cm, find the perimeter of ∆QRS.
2
10
2
8
10
6
6
Monday, May 21, 2012

36. Example 5
Some round cookies are marketed in a triangular package
to pique the consumer’s interest. If ∆QRS is
circumscribed with ⊙T with NQ = 2 cm, QR = 8 cm, and
SM = 10 cm, find the perimeter of ∆QRS.
2
P = 10 + 10 + 2 + 2 + 6 + 6 10
2
8
10
6
6
Monday, May 21, 2012

37. Example 5
Some round cookies are marketed in a triangular package
to pique the consumer’s interest. If ∆QRS is
circumscribed with ⊙T with NQ = 2 cm, QR = 8 cm, and
SM = 10 cm, find the perimeter of ∆QRS.
2
P = 10 + 10 + 2 + 2 + 6 + 6 10
2
P = 36 cm 8
10
6
6
Monday, May 21, 2012

38. Check Your Understanding
p. 721 #1-8
Monday, May 21, 2012

39. Problem Set
Monday, May 21, 2012

40. Problem Set
p. 722 #9-31 odd, 45, 49, 55
“Stop thinking what's good for you, and start thinking
what's good for everyone else. It changes the game.”
- Stephen Basilone & Annie Mebane
Monday, May 21, 2012