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Section 10-5
Tangents
Monday, May 21, 2012

Essential Questions
• How do you use properties of tangents?
• How do you solve problems involving
circumscribed polygons?

Vocabulary
1. Tangent:
2. Point of Tangency:
3. Common Tangent:

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:

Vocabulary
2. Point of Tangency: The point that a tangent
intersects a circle
3. Common Tangent:

Vocabulary
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

Theorems
Theorem 10.10 - Tangents:
Theorem 11.11 - Two Segments:

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:

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

Example 1
Draw in the common tangents for each ﬁgure. If no tangent
exists, state no common tangent.
a. b.

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

Example 2
KL is a radius of ⊙K. Determine whether LM is tangent to
⊙K. Justify your answer.

Example 2
a2
+ b2
= c2

Example 2
a2
+ b2
= c2
202
+ 212
= 292

Example 2
a2
+ b2
= c2
202
+ 212
= 292
400 + 441 = 841

Example 2
a2
+ b2
= c2
202
+ 212
= 292
400 + 441 = 841
Since the values hold for a right triangle,
we know that LM is perpendicular to
KL, making it a tangent.

Example 3
In the ﬁgure, WE is tangent to ⊙D at W. Find the value of x.

Example 3
a2
+ b2
= c2

Example 3
a2
+ b2
= c2
x2
+ 242
= (x + 16)2

Example 3
a2
+ b2
= c2
x2
+ 242
= (x + 16)2
x2
+ 576 = x2
+ 32x + 256

Example 3
a2
+ b2
= c2
x2
+ 242
= (x + 16)2
x2
+ 576 = x2
+ 32x + 256
−x2
−x2

Example 3
a2
+ b2
= c2
x2
+ 242
= (x + 16)2
x2
+ 576 = x2
+ 32x + 256
−x2
−x2
−256−256

Example 3
a2
+ b2
= c2
x2
+ 242
= (x + 16)2
x2
+ 576 = x2
+ 32x + 256
−x2
−x2
−256−256
320 = 32x

Example 3
a2
+ b2
= c2
x2
+ 242
= (x + 16)2
x2
+ 576 = x2
+ 32x + 256
−x2
−x2
−256−256
320 = 32x
x = 10

Example 4
AC and BC are tangent to ⊙Z. Find the value of x.

Example 4
3x + 2 = 4x − 3

Example 4
3x + 2 = 4x − 3
x = 5

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.

Example 5
2
8
10

Example 5
2
8
10
10

Example 5
2
8
10
10
2

Example 5
2
8
10
10
2
6

Example 5
2
8
10
10
2
6
6

Example 5
2
8
10
10
2
6
6
P = 10 + 10 + 2 + 2 + 6 + 6

Example 5
2
8
10
10
2
6
6
P = 10 + 10 + 2 + 2 + 6 + 6
P = 36 cm

Check Your Understanding
p. 721 #1-8

Problem Set

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

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