geometry section 10-5 1112
DESCRIPTION
TangentsTRANSCRIPT
Section 10-5Tangents
Monday, May 21, 2012
Essential Questions
• How do you use properties of tangents?
• How do you solve problems involving circumscribed polygons?
Monday, May 21, 2012
Vocabulary1. Tangent:
2. Point of Tangency:
3. Common Tangent:
Monday, May 21, 2012
Vocabulary1. 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
Vocabulary1. 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
Vocabulary1. 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
TheoremsTheorem 10.10 - Tangents:
Theorem 11.11 - Two Segments:
Monday, May 21, 2012
TheoremsTheorem 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
TheoremsTheorem 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
Example 1Draw in the common tangents for each figure. If no tangent
exists, state no common tangent.
a. b.
Monday, May 21, 2012
Example 1Draw in the common tangents for each figure. If no tangent
exists, state no common tangent.
a. b.
No common tangent
Monday, May 21, 2012
Example 1Draw in the common tangents for each figure. If no tangent
exists, state no common tangent.
a. b.
No common tangent
Monday, May 21, 2012
Example 1Draw in the common tangents for each figure. If no tangent
exists, state no common tangent.
a. b.
No common tangent
Monday, May 21, 2012
Example 2KL is a radius of ⊙K. Determine whether LM is tangent to
⊙K. Justify your answer.
Monday, May 21, 2012
Example 2
a2 + b2 = c2
KL is a radius of ⊙K. Determine whether LM is tangent to ⊙K. Justify your answer.
Monday, May 21, 2012
Example 2
a2 + b2 = c2
202 + 212 = 292
KL is a radius of ⊙K. Determine whether LM is tangent to ⊙K. Justify your answer.
Monday, May 21, 2012
Example 2
a2 + b2 = c2
202 + 212 = 292
400+ 441 = 841
KL is a radius of ⊙K. Determine whether LM is tangent to ⊙K. Justify your answer.
Monday, May 21, 2012
Example 2
a2 + b2 = c2
202 + 212 = 292
400+ 441 = 841
KL is a radius of ⊙K. Determine whether LM is tangent to ⊙K. Justify your answer.
Since the values hold for a right triangle, we know that LM is perpendicular to
KL, making it a tangent.
Monday, May 21, 2012
Example 3In the figure, WE is tangent to ⊙D at W. Find the value of x.
Monday, May 21, 2012
Example 3In the figure, WE is tangent to ⊙D at W. Find the value of x.
a2 + b2 = c2
Monday, May 21, 2012
Example 3In the figure, WE is tangent to ⊙D at W. Find the value of x.
a2 + b2 = c2
x2 + 242 = ( x + 16)2
Monday, May 21, 2012
Example 3In the figure, WE is tangent to ⊙D at W. Find the value of x.
a2 + b2 = c2
x2 + 242 = ( x + 16)2
x2 + 576 = x2 + 32x + 256
Monday, May 21, 2012
Example 3In the figure, WE is tangent to ⊙D at W. Find the value of x.
a2 + b2 = c2
x2 + 242 = ( x + 16)2
x2 + 576 = x2 + 32x + 256
−x2 −x2
Monday, May 21, 2012
Example 3In the figure, WE is tangent to ⊙D at W. Find the value of x.
a2 + b2 = c2
x2 + 242 = ( x + 16)2
x2 + 576 = x2 + 32x + 256
−x2 −x2
−256 −256
Monday, May 21, 2012
Example 3In the figure, WE is tangent to ⊙D at W. Find the value of x.
a2 + b2 = c2
x2 + 242 = ( x + 16)2
x2 + 576 = x2 + 32x + 256
−x2 −x2
−256 −256
320 = 32x
Monday, May 21, 2012
Example 3In the figure, WE is tangent to ⊙D at W. Find the value of x.
a2 + b2 = c2
x2 + 242 = ( x + 16)2
x2 + 576 = x2 + 32x + 256
−x2 −x2
−256 −256
320 = 32x x = 10
Monday, May 21, 2012
Example 4AC and BC are tangent to ⊙Z. Find the value of x.
Monday, May 21, 2012
Example 4AC and BC are tangent to ⊙Z. Find the value of x.
3x + 2 = 4x − 3
Monday, May 21, 2012
Example 4AC and BC are tangent to ⊙Z. Find the value of x.
3x + 2 = 4x − 3
x = 5
Monday, May 21, 2012
Example 5Some 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
Example 5Some 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
810
Monday, May 21, 2012
Example 5Some 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
810
10
Monday, May 21, 2012
Example 5Some 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
810
10 2
Monday, May 21, 2012
Example 5Some 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
810
10 2
6
Monday, May 21, 2012
Example 5Some 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
810
10 2
66
Monday, May 21, 2012
Example 5Some 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
810
10 2
66
P = 10 + 10 + 2 + 2 + 6 + 6
Monday, May 21, 2012
Example 5Some 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
810
10 2
66
P = 10 + 10 + 2 + 2 + 6 + 6
P = 36 cm
Monday, May 21, 2012
Check Your Understanding
p. 721 #1-8
Monday, May 21, 2012
Problem Set
Monday, May 21, 2012
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 MebaneMonday, May 21, 2012