Geometry
Goals
 Know properties of circles.
 Identify special lines in a circle.
 Solve problems with special lines.
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Circle: Set of points on a plane
equidistant from a point (center).
B
This is circle C, or
C
C
AB is a diameter.
R
A
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CR is a radius.
The diameter is twice the radius.
Terminology
 One radius
 Two radii
 radii = ray-dee-eye
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All Radii in a circle are congruent
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Interior/Exterior
A
A is in the interior of the circle.
C
B
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C is on the
circle.
B is in the
exterior of the
circle.
Congruent Circles
Radii are congruent.
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Chord
A chord is a segment between two
points on a circle.
A diameter
is a chord
that passes
through
the center.
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Secant
A secant is
a line that
intersects a
circle at two
points.
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Tangent
•A tangent is a line
that intersects a circle
at only one point.
•It is called the point
of tangency.
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Tangent Circles
Intersect at exactly one point.
These circles are externally tangent.
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Tangent Circles
Intersect at exactly one point.
These circles are internally tangent.
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Can circles intersect at two points?
YES!
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Concentric Circles
Have the same center, different radius.
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Concentric Circles
Have the same center, different radius.
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Concentric Circles
Have the same center, different radius.
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Concentric Circles
Have the same center, different radius.
April 13, 2015
Concentric Circles
Have the same center, different radius.
April 13, 2015
Concentric Circles
Have the same center, different radius.
April 13, 2015
Concentric Circles
Have the same center, different radius.
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Concentric Circles
Have the same center, different radius.
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Common External Tangents
And this is a common external tangent.
This is a common external tangent.
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Common External Tangents in a
real application…
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Common Internal Tangents
And this is a common internal tangent.
This is a common internal tangent.
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Theorem 12.1
(w/o proof)
If a line is tangent to a circle, then it is
perpendicular to the radius drawn to the point of
tangency.
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Theorem 12.2
(w/o proof)
If a line drawn to a circle is perpendicular to a
radius, then the line is a tangent to the circle.
(The converse of 10.1)
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Example 1
Is RA tangent to T?
R
12
5
13
T
A
YES
52 + 122 = 132
25 + 144 = 169
TA = 13
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169 = 169
RAT is a right triangle.
FOIL
Find (x + 3)2
(x + 3)(x + 3)
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FOIL
Find (x + 3)2
(x + 3)(x + 3)
x2
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FOIL
Find (x + 3)2
3x
(x + 3)(x + 3)
x2
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FOIL
Find (x + 3)2
(x + 3)(x + 3)
3x
x2 + 3x
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FOIL
Find (x + 3)2
(x + 3)(x + 3)
9
x2 + 3x + 3x
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FOIL
Find (x + 3)2
(x + 3)(x + 3)
x2 + 3x + 3x + 9
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FOIL
(x + 3)2 = x2 + 6x + 9
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Expand (x + 9)2
 (x + 9)(x + 9)
 F: x2
 O: 9x
 I: 9x
 L: 81
 (x + 9)2 = x2 + 18x + 81
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BC is tangent to circle
A at B. Find r.
Example 2
A
r
AC = r? + 16
D
16
r
B
24
C
DC = 16
r2 + 242 = (r + 16)2
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Solve the equation.
r2 + 242 = (r + 16)2
r2 + 576 = (r + 16)(r + 16)
r2 + 576 = r2 + 16r + 16r + 256
576 = 32r + 256
320 = 32r r2 + 242 = (r + 16)2
r = 10
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Here’s where the situation is now.
A 10
26
D
16
10
B
AC = 26
r = 10
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24
Check: C
102 + 242 = 262
100 + 576 = 676
676 = 676
Theorem 12.3
 If two segments from the same
exterior point are tangent to a circle,
then the segments are congruent.
Theorem Demo
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Example 3
HE and HA are tangent to the circle.
Solve for x.
A
12x + 15
H
9x + 45
E
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Solution
12x + 15 = 9x + 45
3x + 15 = 45
12(10) + 15
A
120 + 15 = 135
12x + 15
3x = 30
H
x = 10
E
9x + 45
9(10) + 45
90 + 45 = 135
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Try This:
The circle is tangent to each side of ABC. Find
the perimeter of ABC.
7 + 12 + 9 = 28
A
2
2
9
7
7
C
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5
7
5
12
B
Can you…
 Identify a radius, diameter?
 Recognize a tangent or secant?
 Define Concentric circles? Internally
tangent circles? Externally tangent?
 Tell the difference between internal
and external tangents?
 Solve problems using tangent
properties?
April 13, 2015
Practice Problem 1
MD and ME are tangent to the circle.
Solve for x.
4x – 12 = 2x + 12
D
4x  12
2x – 12 = 12
M
2x = 24
x = 12
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2x + 12
E
Practice Problem 2
R
x
4
T
Solve for x.
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12
x2 + 42 = (4 + 12)2
x2 + 16 = 256
x2 = 240
x = 415  15.5
Practice Problem 3
R
8
x
T
x
6
x2 + 82 = (x + 6)2
x2 + 64 = x2 + 12x + 36
64 = 12x + 36
Solve for x.
28 = 12x
x = 2.333…
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Practice Problems
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