Circles
A circle is the set of
all points in a plane
that are equidistant
from a given point
called the center of
the circle.
circle A, or A
A
In a plane, the interior of a circle
consists of the points that are inside
the circle. The exterior of a circle
consists of the points that are outside
the circle.
interior
exterior
A
Segments
and Lines
The segment joining the
center to a point on the
circle is the radius of
the circle.
Example: AG or AH
Two circles are
congruent if they have
the same radius
measures.
A
H
G
A chord is a
segment whose
endpoints are points
on the circle.
Ex: CD or EF
D
C
A
E
F
A diameter is a
chord that passes
through the center
of the circle.
Ex: BC
B
A
C
The distance across
the circle, through its
center, is the measure
of the diameter of the
circle.
Ex: BC
B
A
C
The length of a
diameter is twice the
length of a radius. The
length of a radius is
half the length of a
diameter.
B
A
C
A secant is a line
that intersects a
circle in two points
A tangent is a line in the plane of a
circle that intersects the circle in exactly
one point. The point at which a tangent
line intersects the circle to which it is
tangent is the point of tangency.
In a plane, two circles can intersect in
two, one or no points. Coplanar
circles that intersect in one point are
called tangent circles.
Coplanar circles
that have a
common center
are called
concentric.
A
A line or segment that is tangent to
two coplanar circles is called a
common tangent.
A common internal tangent intersects
the segment that joins the centers of
the two circles.
A common external tangent does not
intersect the segment that joins the
centers of the two circles.
Theorems
 If a line is tangent to a circle, then it is
perpendicular to the radius drawn to the
point of tangency.
 In a plane, if a line is perpendicular to a
radius of a circle at its endpoint on the
circle, then the line is tangent to the circle.
Theorem
If two segments from the same exterior
point are tangent to a circle, then they are
congruent.
Angles
and Arcs
In a plane, an angle whose vertex
is the center of a circle is a central
angle of the circle.
B
central
angle
A
C
If the measure of a central angle is
less than 180˚, then B and C and the
points of A in the interior of BAC
form a minor arc.
B
minor arc
A
C
The measure of a minor arc is
defined to be the measure of its
central angle.
B
A
x
x
C
If the two endpoints of an arc are the
endpoints of a diameter, then the
arc is a semicircle.
B
D
A
C
semicircle
The measure of a semicircle is 180˚.
B
D
A
C
semicircle
If the measure of a central angle is
greater than 180˚, then B and C and
the points of A in the exterior of
BAC form a major arc of the circle.
B
major arc
A
C
The measure of a major arc is
defined as the difference between
360˚ and the measure of its
associated minor arc.
B
(360 – x)
A
x
C
Arc Addition Postulate
The measure of an arc formed by
two adjacent arcs is the sum of the
measures of the two arcs.
B
mBD  mDC  mBDC
D
A
C
Inscribed
Angles
An inscribed angle is an angle whose
vertex is on a circle and whose sides
contain chords of the circle. The arc that
lies in the interior of an inscribed angle
and has endpoints on the angle is called
the intercepted arc of the angle.
Measure of an Inscribed Angle
If an angle is inscribed
in a circle, then its
measure is half the
measure of its
intercepted arc.
Theorem
If two inscribed angles
of a circle intercept the
same arc, then the
angles are congruent.
Properties of Inscribed Polygons
If all the vertices of a
polygon lie on a circle,
the polygon is inscribed
in the circle and the
circle is circumscribed
about the polygon.
Theorems About Inscribed
Polygons
If a right triangle is
inscribed in a circle,
then the hypotenuse
is a diameter of the
circle.
Theorems About Inscribed
Polygons
Conversely, if one side
of an inscribed triangle
is a diameter of the
circle, then the triangle
is a right triangle and
the angle opposite the
diameter is the right
angle.
A quadrilateral can be inscribed in a
circle if and only if its opposite
angles are supplementary.
Other Angle
Relationships in
Circles
Theorem about Chords of Circles
In the same circle, or in congruent
circles, two minor arcs are congruent
if and only if their corresponding
chords are congruent.
Theorem about Chords of Circles
If a diameter of a circle
is perpendicular to a
chord, then the
diameter bisects the
chord and its arc.
Theorem about Chords of Circles
If one chord is a
perpendicular bisector
of another chord, then
the first chord is a
diameter.
Theorem about Chords of Circles
In the same circle, or
in congruent circles,
two chords are
congruent if and only if
they are equidistant
from the center.
Theorem
If a tangent and a
secant intersect at a
point on a circle, then
the measure of each
angle is one half the
measure of its
intercepted arc.
If two chords intersect
in the interior of a circle,
then the measure of
each angle is one half
the sum of the
measures of the arcs
intercepted by the angle
and its vertical angle.
C
D
F
B
A
E
mCFB  1 mCB  mDE
2




mDFE  1 mCB  mDE
2
C
D
F
B
A
E
mCFD  1 mCD  mBE
2




mBFE  1 mCD  mBE
2
C
D
F
B
A
E
If a tangent and a secant intersect in
the exterior of a circle, then the
measure of the angle formed is one
half the difference of the measures
of the intercepted arcs.

mBVA  1 mBA  mCA
2

If two tangents intersect in the
exterior of a circle, then the measure
of the angle formed is one half the
difference of the measures of the
intercepted arcs.

mBVA  1 mBCA  mBA
2

If two secants intersect in the
exterior of a circle, then the measure
of the angle formed is one half the
difference of the measures of the
intercepted arcs.

mBVA  1 mBA  mDC
2

Segment Lengths
in Circles
If two chords intersect in the interior
of a circle, then the product of the
lengths of the segments of one
chord is equal to the product of the
lengths of the segments of the other
chord.
Ex: 8x = 24
x=3
8 × 3 = 4 × 6
If two secant segments share the same
endpoint outside a circle, then the product
of the length of one secant segment and
the length of its external segment equals
the product of the length of the other
secant segment and the length of its
external segment.
outside segment × whole segment = outside segment × whole segment
Ex: 4(x + 4) = 3(3 + 5)
4x + 16 = 24
4x = 8
x=2
4(2 + 4) = 3 (3 + 5)
If a secant segment and a tangent
segment share an endpoint outside a
circle, then the product of the length of
the secant segment and the length of its
external segment equals the square of
the length of the tangent segment.
tangent squared = outside segment × whole segment
Ex: 62 = 4(4 + x)
36 = 16 + 4x
20 = 4x
5=x
62 = 4(4 + 5)
Equations of
Circles
You can write an equation of a
circle in a coordinate plane if you
know its radius and the
coordinate of its center.
Suppose the radius of a circle is r
and the center is (h, k). Let (x, y) be
any point on the circle.
The distance between (x, y) and (h, k)
is r, so you can use the
Distance Formula.
r  ( x  h)  ( y  k )
2
2
Square both sides to find the
standard equation of a circle with
radius r and center (h, k).
(x – h)2 + (y – k)2 = r 2
If the center is the origin, then the
standard equation is x 2 + y 2 = r 2