tisk & 2 mm lesson 9-5: tangents homework: 9-5 problems in packet 2 monday, february 11, 2013...
TRANSCRIPT
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• TISK & 2 MM• Lesson 9-5: Tangents• Homework: 9-5 problems in packet 2
Monday, February 11, 2013Agenda
TISK Problems
1. Simplify completely:
2. Write the equation of a line in slope-intercept form that passes through the point (3,-8) and is perpendicular to the line .
3. Name an angle congruent to angle 3; state theorems or postulates that justify your answer. 3
1
4
2
5 6
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§9-5 Tangents
Definitions Secant Tangent
A secant is a line that passes through a circle, intersecting
the circle twice.
A tangent is a line that passes through a circle,
intersecting the circle exactly once.
The point where the line intersects the circle is called the point of tangency.
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More Definitions
Definitions Tangent Circles
Two circles are tangent if they intersect in only one point.
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More Definitions
Definitions Tangent Circles Common
Tangent Internal External
F
A
E
BDC
Two circles are tangent if they intersect in only one point.
Common internal tangents are “between” the two circles.
Common external tangents are “outside” the two circles.
is a common internal tangent
is an common external tangent
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Theorems
If a line is tangent to a circle, then it is perpendicular to the radius drawn to the point of tangency.
l
Q
P
If is tangent to at Q P then PQ
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Theorems
In a plane, if a line is perpendicular to a radius of a circle at its endpoint on the circle, then the line is tangent to the circle.
l
Q
P
then is tangent to at Q P If at PQ P
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Theorems
Theorem If two segments from the same exterior point are
tangent to a circle, then they are congruent.
DA
B
C If and are tangent to CD BD A���������������������������������������� ���
then CD BD
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Example 1. Find the value of x.
AC
B
D
is tangent to at AB C B�������������� �
is tangent to at AD C D
�������������� �
2 2x
11
2 2x
11
2 2
2 9x3x
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Example 2. Find the value of x.
A C
B
D
is tangent to at AB C B�������������� �
is tangent to at AD C D
�������������� �
2 8x
44
2 8x
44
8 8
2 36x6x
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Your Turn:
Find the values of x, y, and z in the diagram.
15
y
36
z
x
A
C
B
D
15y
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Your Turn:
Find the values of x, y, and z in the diagram.
15
y
36
z
x
A
C
B
D
36z
15y
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15
y
36
z
x
A
C
B
D
Your Turn:
Find the values of x, y, and z in the diagram.
22 236y AC
22 215 36 AC
2 2 23 5 3 12 AC
3 13 39AC
39 15x15y
24x
36z
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Prove it!
H
F
D
B
A
C
E
Given: is tangent to at CH B C
�������������� �
is tangent to at CH E H�������������� �
is tangent to at AF B A�������������� �
is tangent to at AF E F�������������� �
Prove: CH AF
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Prove it!
H
F
D
B
A
C
E
Given: is tangent to at CH B C
�������������� �
is tangent to at CH E H�������������� �
is tangent to at AF B A�������������� �
is tangent to at AF E F�������������� �
Prove: CH AF
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Prove it!
H
F
D
B
A
C
E
Given: is tangent to at CH B C
�������������� �
is tangent to at CH E H�������������� �
is tangent to at AF B A�������������� �
is tangent to at AF E F�������������� �
Prove: CH AF
CD AD
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Prove it!
H
F
D
B
A
C
E
Given: is tangent to at CH B C
�������������� �
is tangent to at CH E H�������������� �
is tangent to at AF B A�������������� �
is tangent to at AF E F�������������� �
Prove: CH AF
CD AD
DF DH
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Prove it!
H
F
D
B
A
C
E
Given: is tangent to at CH B C
�������������� �
is tangent to at CH E H�������������� �
is tangent to at AF B A�������������� �
is tangent to at AF E F�������������� �
Prove: CH AF
DF DH
CD AD
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Prove it!
H
F
D
B
A
C
E
Given: is tangent to at CH B C
�������������� �
is tangent to at CH E H�������������� �
is tangent to at AF B A�������������� �
is tangent to at AF E F�������������� �
Prove: CH AF
CD ADDF DH
CH CD DH
AF AD DF
CH AD DF
CH AFCH AF
DF DH
CD AD