10.1 The Circle
After studying this section, you will be
able to identify the characteristics of
circles, recognize chords, diameters,
and special relationships between radii
and chords.
Basic Properties and Definitions
Definition
A circle is the set of all points in a plane
that are a given distance from a given point
in the plane. The given point is the center
of the circle, and the given distance is the
radius. A segment that joins the center to
a point on the circle is also called a radius.
(The plural of radius is radii).
center
circle
Definition
Two or more coplanar circles with the
same center are called concentric circles.
Definition
Two circle are congruent if they have
congruent radii.
5
5
Definition
A point is inside (in the interior of) a circle if
its distance from the center is less than the
radius. Points A and B are in the interior of Circle B
Definition
A point is outside (in the exterior of) a
circle if its distance from the center is
greater than the radius.
Point C is in the exterior of Circle B
D
Definition
A
B
center is
A point is on a circle if its
distance from the
equal to the radius.
Point D is on Circle B
C
Chords and Diameters
Points on a circle can be connected
by segments called chords
Definition
A chord of a circle is a segment joining any
two points on the circle.
Definition
A diameter of a circle is
a chord that passes
through the center of
the circle.
Circumference and Area of a
Circle
Area of a Circle
A r
2
Circumference of a Circle
C  2r or C   d
where r is the radius of the circle and
  3.14
Radius-Chord Relationships
OP is the distance from O to chord AB
A
P
O
Definition
B
The distance from the center of a circle to
a chord is the measure of the
perpendicular segment from the center to
the chord
Theorems
Theorem
If a radius is perpendicular to a chord, this
it bisects the chord.
A
D
E
O
B
Theorems
Theorem
If a radius of a circle bisects a chord that is
not a diameter, then it is perpendicular to
that chord.
E
H
G
O
F
Theorems
Theorem
The perpendicular bisector of a chord
passes through the center of the circle.
C
P
Q
O
D
Example 1
Given: Circle Q
PR  ST
Prove PS  PT
S
P
Q
R
T
Example 2
The radius of circle O is 13 mm. The length of
chord PQ is 10 mm. Find the distance from
chord PQ to the center, O
P
O
Q
Example 3
Given: Triangle ABC is isosceles
Similar circles P and Q
 AB  AC 
BC PQ
Prove: Circle Q  Circle P
A
B
P
C
Q
Summary
Explain how to determine
whether a point is on a circle, in
the interior of a circle, or in the
exterior of a circle.
Homework: worksheet