Download Chapter 10 P3

Chapter 10
CIRCLES
Ms. Watson
Geometry
Banneker Academic High School
GOAL #1
Identify segments and lines
related to circles.
Who’s N’ The Circle Fam?
VOCABULARY
DEFINITION
CIRCLE
-the set of all points in a plane that are
equidistant from a given point called the
center of a circle
RADIUS
-the distance from the center to a point on
the circle
DIAMETER
-the distance across the circle, through its
center (the diameter is 2x the radius)
radius
diameter
Who’s N’ The Circle Fam?
VOCABULARY
DEFINITION
-a segment whose endpoints are points on
the circle PS and PR are chords
CHORD
-a chord that passes through the center of the
circle
DIAMETER
-a line that intersects a circle in two points
SECANT
TANGENT
-a line in the plane of a circle that intersects
the circle in exactly one point
secant
R
Q
tangent
P
S
Who’s N’ The Circle Fam?
Common External Tangent
Common Tangents
A line or line segment
that is tangent to
two circles in the
same plane is called
a common tangent.
2 Types
Common Internal Tangent
GOAL #2
Use properties of tangent
to a circle.
Theorem 10.1
If a line is tangent to a
circle, then it is
perpendicular to the
radius drawn to the
point of tangency.
What do we know about right triangles???
How can we use what we know to solve the length of sides of a triangle???
Theorem 10.3
If two segments from
the same exterior
point are tangent to
a circle, then they
are congruent
(equal).
10.2
Arcs & Chords
GOAL #1
Use properties of arcs of
circles.
Who’s N’ The Circle Fam?
VOCABULARY
DEFINITION
-an angle whose vertex is the center of a
circle
CENTRAL ANGLE
MINOR ARC
MAJOR ARC
-part of a circle that measures less than 180°
SEMICIRCLE
-an arc whose endpoints are the end points of
a diameter of the circle
-part of a circle that measures between 180°
and 360°
A
Minor Arc
C
K
Major Arc
B
Name That Arc!
Arcs are named by their endpoints.
MINOR ARCS
MAJOR ARCS & SEMICIRCLES
-named by their endpoints
-the minor arc associated with  AKC is AC
-named by their endpoints and by a
point on the arc
-the major arc associated with  AKC
is ABC
A
Minor Arc
C
K
Major Arc
B
Measure That Arc!
MINOR ARCS
MAJOR ARCS & SEMICIRCLES
-the same as the measure of its central angle -the difference between 360° and the
measure of its associated minor arc
- 60 °
- 360° - 60 ° = 300 °
A
60 °
300 °
Major Arc
Minor Arc
60 °
C
K
B
Arc Addition Postulate
Discovery
Step 1:
Draw a circle.
Step 2:
Place three points on the circle
named A, B, and C.
Discover that…
The measure of an arc formed by two adjacent
arcs is the sum of the measures of the two
arcs.
mABC = mAB + mBC
Arc for Thought
• Two arcs of the same circle or of congruent
circles are congruent arcs if they have the
same measure. So, two minor arcs of the
same circle or of congruent circles are
congruent if their central angles are
congruent.
Which minor arcs are congruent?
Why are they congruent?
GOAL #2
Use properties of chords of
circles.
Theorem 10.4
A
C
B
Theorem 10.5
Theorem 10.6
If one chord is a perpendicular
bisector of another chord, the first
chord is a diameter.
J
M
K
L
Theorem 10.7
Hands on Activity
10.3
Inscribed Angles
GOAL #1
Use inscribed angles to
solve problems.
Who’s N’ The Circle Fam?
Inscribed Angle
-an angle whose vertex is
on a circle and whose
sides contain chords of
the circle
Intercepted Arc
-an arc that lies in the
interior or an inscribed
angle and has endpoints
on the angle
Theorem 10.8
Measure of an Inscribed Angle
If an angle is inscribed in a circle,
then its measure is half the
measure of its intercepted arc.
1
m  ABC = mAC
A
C
2
B
Theorem 10.9
If two inscribed angles
of a circle intercept the
same are, then the
angles are congruent.
A
C
B
C
D

D
GOAL #2
Use properties of inscribed
polugons.
Who’s N’ The Circle Fam?
If all of the vertices of a polygon lie on a
circle, the polygon is INSCRIBED in the
circle and the circle is CIRCUMSCRIBED
about the polygon.
Theorem 10.10
If a right triangle is
inscribed in a circle,
then the hypotenuse is
a diameter of the
circle. Also the angle
opposite the diameter A
is a right angle.
B
 B is a right angle if and
only if AC is a diameter
of the circle.
C
Theorem 10.11
A quadrilateral can be
inscribed in a circle if
and only if its opposite
angles are
supplementary.
E
D, E, F, and G lie on the circle if
and only if
F
G
D
m D + m F = 180
And
m E + m G = 180

