Transcript Geometry - BakerMath.org

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.
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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.
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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?
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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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