geometry
DESCRIPTION
Geometry. Circles 10.1. Goals. Know properties of circles. Identify special lines in a circle. Solve problems with special lines. CR is a radius. AB is a diameter. Circle: Set of points on a plane equidistant from a point (center). B. This is circle C, or. C. R. A. - PowerPoint PPT PresentationTRANSCRIPT
April 19, 2023
Goals
Know properties of circles. Identify special lines in a circle. Solve problems with special lines.
April 19, 2023
Circle: Set of points on a plane equidistant from a point (center).
C
This is circle C, or
C
R
CR is a radius.
A
B
AB is a diameter.
The diameter is twice the radius.
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Interior/Exterior
AA is in the interior of the circle.
B
B is in the exterior of the circle.
C
C is on the circle.
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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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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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Common External Tangents
This is a common external tangent.
And this is a common external tangent.
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Common Internal Tangents
This is a common internal tangent.
And 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
R A
T
5
12
13
YES
TA = 13RAT is a right triangle.
52 + 122 = 132
25 + 144 = 169
169 = 169
Is RA tangent to T?
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Example 2BC is tangent to circle A at B. Find r.
A
B C
r16
24
D
DC = 16
rAC = ?AC = r + 16
r2 + 242 = (r + 16)2
April 19, 2023
r2 + 242 = (r + 16)2
r2 + 576 = (r + 16)(r + 16)
r2 + 576 = r2 + 16r + 16r + 256
576 = 32r + 256
320 = 32r
r = 10
Solve the equation.
r2 + 242 = (r + 16)2
April 19, 2023
A
B C
1016
24
D
AC = 26
10
Here’s where the situation is now.
Check:
102 + 242 = 262
100 + 576 = 676
676 = 676
26
r = 10
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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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Solution
12x + 15 = 9x + 45
3x + 15 = 45
3x = 30
x = 10H
A
E
12x + 15
9x + 459(10) + 45
90 + 45 = 135
12(10) + 15
120 + 15 = 135
April 19, 2023
Try This:
The circle is tangent to each side of ABC. Find the perimeter of ABC.
A
BC
7
2
5
2
57
9 7
12
7 + 12 + 9 = 28
April 19, 2023
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 19, 2023
Practice Problem 1
MD and ME are tangent to the circle. Solve for x.
D
E
M
4x 12
2x + 12
4x – 12 = 2x + 12
2x – 12 = 12
2x = 24
x = 12
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Practice Problem 2
x2 + 42 = (4 + 12)2
x2 + 16 = 256
x2 = 240
x = 415 15.5
R
T
4
12
x
Solve for x.
April 19, 2023
Practice Problem 3
x2 + 82 = (x + 6)2
x2 + 64 = x2 + 12x + 36
64 = 12x + 36
28 = 12x
x = 2.333…
R
T6
8
x
Solve for x.
x