geometry june 8, 2015geometry 10.1 tangents to circles2 goals know properties of circles. identify...
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Geometry
April 18, 2023 Geometry 10.1 Tangents to Circles 2
Goals
Know properties of circles. Identify special lines in a circle. Solve problems with special lines.
April 18, 2023 Geometry 10.1 Tangents to Circles 3
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.
April 18, 2023 Geometry 10.1 Tangents to Circles 4
Terminology
One radius Two radii radii = ray-dee-eye
April 18, 2023 Geometry 10.1 Tangents to Circles 5
All Radii in a circle are congruent
April 18, 2023 Geometry 10.1 Tangents to Circles 6
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.
April 18, 2023 Geometry 10.1 Tangents to Circles 7
Congruent Circles
Radii are congruent.
April 18, 2023 Geometry 10.1 Tangents to Circles 8
April 18, 2023 Geometry 10.1 Tangents to Circles 9
Chord
A chord is a segment between two points on a circle.
A diameter is a chord that passes through the center.
April 18, 2023 Geometry 10.1 Tangents to Circles 10
Secant
A secant is a line that intersects a circle at two points.
April 18, 2023 Geometry 10.1 Tangents to Circles 11
Tangent
•A tangent is a line that intersects a circle at only one point.
•It is called the point of tangency.
April 18, 2023 Geometry 10.1 Tangents to Circles 12
Tangent Circles
Intersect at exactly one point.
These circles are externally tangent.
April 18, 2023 Geometry 10.1 Tangents to Circles 13
Tangent Circles
Intersect at exactly one point.
These circles are internally tangent.
April 18, 2023 Geometry 10.1 Tangents to Circles 14
Can circles intersect at two points?
YES!
April 18, 2023 Geometry 10.1 Tangents to Circles 15
Concentric Circles
Have the same center, different radius.
April 18, 2023 Geometry 10.1 Tangents to Circles 16
Concentric Circles
Have the same center, different radius.
April 18, 2023 Geometry 10.1 Tangents to Circles 17
Concentric Circles
Have the same center, different radius.
April 18, 2023 Geometry 10.1 Tangents to Circles 18
Concentric Circles
Have the same center, different radius.
April 18, 2023 Geometry 10.1 Tangents to Circles 19
Concentric Circles
Have the same center, different radius.
April 18, 2023 Geometry 10.1 Tangents to Circles 20
Concentric Circles
Have the same center, different radius.
April 18, 2023 Geometry 10.1 Tangents to Circles 21
Concentric Circles
Have the same center, different radius.
April 18, 2023 Geometry 10.1 Tangents to Circles 22
Concentric Circles
Have the same center, different radius.
April 18, 2023 Geometry 10.1 Tangents to Circles 23
Common External Tangents
This is a common external tangent.
And this is a common external tangent.
April 18, 2023 Geometry 10.1 Tangents to Circles 24
Common External Tangents in a real application…
April 18, 2023 Geometry 10.1 Tangents to Circles 25
Common Internal Tangents
This is a common internal tangent.
And this is a common internal tangent.
April 18, 2023 Geometry 10.1 Tangents to Circles 26
April 18, 2023 Geometry 10.1 Tangents to Circles 27
Theorem 10.1 (w/o proof)
If a line is tangent to a circle, then it is perpendicular to the radius drawn to the point of tangency.
April 18, 2023 Geometry 10.1 Tangents to Circles 28
Theorem 10.2 (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)
April 18, 2023 Geometry 10.1 Tangents to Circles 29
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?
April 18, 2023 Geometry 10.1 Tangents to Circles 30
FOILFind (x + 3)2
(x + 3)(x + 3)
April 18, 2023 Geometry 10.1 Tangents to Circles 31
FOILFind (x + 3)2
(x + 3)(x + 3)
x2
April 18, 2023 Geometry 10.1 Tangents to Circles 32
FOILFind (x + 3)2
(x + 3)(x + 3)
x2
3x
April 18, 2023 Geometry 10.1 Tangents to Circles 33
FOILFind (x + 3)2
(x + 3)(x + 3)
x2 + 3x
3x
April 18, 2023 Geometry 10.1 Tangents to Circles 34
FOILFind (x + 3)2
(x + 3)(x + 3)
9
x2 + 3x + 3x
April 18, 2023 Geometry 10.1 Tangents to Circles 35
FOILFind (x + 3)2
(x + 3)(x + 3)
x2 + 3x + 3x + 9
April 18, 2023 Geometry 10.1 Tangents to Circles 36
FOIL(x + 3)2 = x2 + 6x + 9
April 18, 2023 Geometry 10.1 Tangents to Circles 37
Expand (x + 9)2
(x + 9)(x + 9) F: x2
O: 9x I: 9xL: 81 (x + 9)2 = x2 + 18x + 81
April 18, 2023 Geometry 10.1 Tangents to Circles 38
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 18, 2023 Geometry 10.1 Tangents to Circles 39
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 18, 2023 Geometry 10.1 Tangents to Circles 40
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
April 18, 2023 Geometry 10.1 Tangents to Circles 41
Theorem 10.3
If two segments from the same exterior point are tangent to a circle, then the segments are congruent.
Theorem Demo
April 18, 2023 Geometry 10.1 Tangents to Circles 42
Example 3
HE and HA are tangent to the circle. Solve for x.
H
A
E
12x + 15
9x + 45
April 18, 2023 Geometry 10.1 Tangents to Circles 43
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 18, 2023 Geometry 10.1 Tangents to Circles 44
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 18, 2023 Geometry 10.1 Tangents to Circles 45
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 18, 2023 Geometry 10.1 Tangents to Circles 46
Practice Problems
Skip
April 18, 2023 Geometry 10.1 Tangents to Circles 47
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
April 18, 2023 Geometry 10.1 Tangents to Circles 48
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 18, 2023 Geometry 10.1 Tangents to Circles 49
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