Chapter 10 Properties of Circles. 10.1 Using Properties of Tangents Circle- a set of all points in a plane that are equidistant from a given point called.
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Chapter 10Properties of Circles
10.1 Using Properties of TangentsCircle- a set of all points in a
plane that are equidistant from a given point called the center
C
Radius- a segment whose endpoints are the center and any point on the circle
Chord- a segment whose endpoints are on a circle
Diameter- a chord that contains the center of the circle
Secant- a line that intersects a circle in two points
Tangent- a line in the plane of a circle that intersects the circle in exactly one point
Can you name it?ChordRadiusDiameterSecantTangentPoint of Tangency
C
E
D
B
J
A
G
F
H
I
Coplanar circlesConcentric circles
Internally tangent circles
Externally tangent circles
Common tangentsInternal common tangent
External common tangent
TheoremsIn a plane, a line is tangent to a
circle if and only if the line is perpendicular to a radius of the circle at its endpoint on the circle
mQ
P
Tangent segments from a common external point are congruent.
STRSP
T
S
R
ExamplesIs segment BC tangent to circle A
if segment AB is a radius?
6725
60
A
B C
ExampleS is a point of tangency. Find r.
r36r
48
T
S R
ExamplePoint R and T are tangent to
circle P. Find x.
6x-8
25
P
T
S
R
ExampleHow many common tangents?
B.
A.C.
10.2 Finding Arc MeasuresCentral angle- an angle whose
vertex is the center of the circle
Major arc
Minor arc
Semicircle
80C
A
D
BE
Arc Addition PostulateThe measure of an arc formed by
two adjacent arcs is the sum of the measures of the two arcs.
C
A
D
B
E
Congruent Circles and ArcsTwo circles are congruent if they
have the same radius.Two arcs are congruent if they
have the same measure and they are arcs of the same circle or of congruent circles.
The radii of a circle, or of congruent circles, are congruent.
Examples
Find the following:
AE
ABD
EBD
135C
A
D
B
E
ExampleAre arcs AB and DE congruent?
A.
45
45C
A
D
B
E
B.
E
C
A
D
B
C.
C
A
B
D
E
ExampleAges of people in a town (in
years)
mGA
mGE
mGFE
mBFA60 80
90
30
100C
A
E
G
F
B
10.3 Applying Properties of ChordsIn the same circle, or in
congruent circles, two minor arcs are congruent if and only if their corresponding chords are congruent.
Example: Find the measure of arc SR.
100
P Q
R
S T
U
If one chord is a perpendicular bisector of another chord, then the first chord is a diameter.
If the diameter of a circle is perpendicular to a chord, then the diameter bisects the chord and its arc.
Example
FindBFCFAF
10
12
F
C
A
E
D B
In the same circle, or in congruent circles, two chords are congruent if and only if they are equidistant from the center.
A
B C
D
E
ExampleBC= 2x +6ED = 3x – 1 Find BC
A
B C
D
E
ExampleThree props are placed on a
stage (P,Q,R). Where do you put the table so that it is the same distance from each prop?
P
Q
R
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