Transcript Section 12.1: Lines That intersect Circles

Section 12.1: Lines That
intersect Circles
By: The Balloonicorns
Stuff to learn;
• Identify tangents, secants, and
chords
• Use properties of tangents to
solve problems
Words and Phrases to
Remember
•
Interior of a Circle – The set of all points inside the circle
•
Exterior of a Circle – The set of all points outside the circle
•
Chord – A segment whose endpoints lie on a circle
•
Secant – A line that intersects a circle at two points
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Tangent of a Circle – A line in the same plane as a circle that intersects it at exactly one point
•
Point of Tangency – The point where the tangent and circle intersect
•
Congruent Circles – Two circles that have congruent radii
•
Concentric Circles – Coplanar circles with the same center
•
Tangent Circles – Two coplanar circles that intersect at exactly one point
•
Common Tangent – A line that is tangent to two circles
Example of the Lines and
Segments
Examples of Pairs of
Circles
Concentric Circles
Tangent Circles
Common Tangent
•
14
12
How to Identify Tangents
of Circles
• Center of circle A is (4,
4), and its radius is 4.
The center of circle B is
(5, 4) and its radius is 3.
The two circles have one
point of intersection (8,
4). The vertical line x =
8 is the only common
tangent of the two
circles.
10
8
6
4
A
B
2
5
10
15
20
THEOREMS
• 12-1-1: If a line is tangent to a
circle, then it is perpendicular to
the radius drawn to the point of
tangency.
• 12-1-2: If a line is perpendicular
to a radius of a circle at a point
on the circle, then the line is
tangent to the circle.
• 12-1-3: If two segments are
tangent to a circle from the same
external point, then the segments
are congruent.
B
16 ft.
r
C
8 ft.
c2
=
a2
+
b2
(r + 8)2 = r2 + 162
r 2 + 16r + 64 = r2 + 256
Pythagorean Thm.
A
Substitute values
Square of binomial
16r + 64 = 256
Subtract r2 from each side.
16r = 192
Subtract 64 from each side
r = 12
Divide.
How To use Tangents
r
•
Practice
D
A
C
11
B
•
AB = AD
11 = x2 + 2
Two tangent segments from the same point are 
Substitute values
9 = x2
Subtract 2 from each side.
3=x
Find the square root of 9.
Using Properties of
Tangents
•
Practice