Download Chapter 5: Discovering and Proving Polygon Properties

Ch 6 Short Summary L1 Key
Name ____________________________
Chapter 6: Discovering and Proving Circle Properties
Segments:
Types of Lines [Segments]
Radius [of a circle] is
• A segment that goes from the
center to any point on the circle.
• The distance from the center to
any point on the circle, r.
Diameter [of a circle] is
• A chord that goes through the
center of a circle.
• The length of the diameter. d = 2r
or ½ d = r.
Chord is a segment connecting
any two points on the circle.
Secant A line that intersects a
circle in two points.
Tangent [to a circle] is a line that
intersects a circle in only one
point.
The point of intersection is called
the point of tangency.
Tangent Properties
Tangent ⊥ radius (& visa versa).
JJJG
JJJG
JJJG
JJJG
NA ⊥ AE ; NG ⊥ EG
NA = NG
m∠N = 180 − m∠AEG
p
= 180 − mAG
N
Tangents NA and NG
A
Tangents from a point are
congruent.
E
G
p
m∠N = 180 − mA
G (Think kite with
two 90º angles.)
Chord Properties
If 2 chords are congruent, then
central angles are congruent.
intercepted arcs are congruent.
chords are equidistant from the center.
B
D
O
M
N
A
C
FM = ME
p = mPE
p
mPF
F
O
P
M
E
N
H
B
⊥
bisector of HG , it
G If you draw
passes through O.
If AB & CD then
A
C
D
m∠BOA = m∠COD
p = mDC
p
mAB
MO = ON
If OP ⊥ EF then
Perpendiculars & Chords
Perpendicular from center bisects the
chord and the arc.
Perpendicular bisector of a chord passes
through the center.
Parallel Chords or Secants
Parallel lines [secants or chords]
intercept congruent arcs.
If AB ≅ CD then
p = mBC
p
mAD
Ch 6 Short Summary L1 Key
Name ____________________________
Angles & Arcs:
Arcs & Angles
Vertex?
Formula:
Center
angle = arc
D
Central Angles
central angle An angle whose vertex lies
on the center of a circle and whose sides
are radii of the circle
O
C
A
B
Central angle determines the measure of
an intercepted minor arc.
On
angle = ½ arc
Inside
angle = ½ sum
Outside
angle = ½ difference
p = m∠AOC , in degrees.
mAC
D
Inscribed Angles
inscribed angle An angle whose vertex lies
on a circle and whose sides are chords of
the circle.
Inscribed Angle Conjecture
The measure of an angle inscribed in a
circle equals half the measure of its
intercepted arc.
Inscribed Angles Intercepting Arcs
Conjecture
Inscribed angles that intercept the same
arc [or congruent arcs] are congruent.
O
B
C
80
O
40
A
R
L
100
J
50
50
M
K
Angles Inscribed in a Semicircle
Conjecture
Angles inscribed in a semicircle are
right angles.
C
A
C
90
E
D
Arcs
Arc [of a circle] is formed by two points on
a circle and a continuous part of the circle
between them. The two points are called
endpoints.
Arc Addition Postulate If point B is
AC and between points A and C, then
on p
p
p = mAC
p.
mAB + m BC
Semicircle is an arc whose endpoints are
the endpoints of the diameter.
Minor arc is an arc that is smaller than a
AC
semicircle, p
Major arc is an arc that is larger than a
ADC
semicircle, q
Intercepted Arc An arc that lies in the
interior of an angle with endpoints on the
sides of the angle.
Ch 6 Short Summary L1 Key
Circles
Circle is a set of points in a plane a
given distance (radius) from a given
point (center).
Sphere is a set of points a given
distance (radius) from a given point
(center).
Congruent circles are two or more
circles with the same radius measure.
Concentric circles are two or more
circles with the same center point.
Tangent Circles Circles that are
tangent to the same line at the same
point.
Internally Tangent Circles Two
tangent circles having centers on the
same side of their common tangent.
Externally Tangent Circles Two
tangent circles having centers on
opposite sides of their common
tangent.
Name ____________________________
Circumference & Arc Length
A
O
60
5
B
Circumference
The perimeter of a circle, which is the
distance around the circle.
Also the curved path of the circle itself.
Circumference Conjecture
If C is the circumference, d = diameter,
r = radius:
C = 2rπ or dπ
Measure of an Arc: The measure of an arc
equals the measure of its central angle,
measured in degrees.
VS
Arc length: The portion of (or fraction of)
the circumference of the circle described
by an arc, measured in units of length.
Arc Length Conjecture
Arc length = a fraction of a circle
Arc length =
m arc
•C
360
Ch 6 Short Summary L1 Key
Quadrilaterals
Cyclic Quadrilateral A quadrilateral that can
be inscribed in a circle. (Each of its angles are
inscribed angles and each of its sides is a
chord of the circle.)
Name ____________________________
A
C
B
D
Cyclic Quadrilateral Conjecture
The opposite angles of a cyclic quadrilateral
are supplementary.
Parallelogram Inscribed in a Circle
Theorem
If a parallelogram is inscribed within a
circle, then the parallelogram is a rectangle.
G
O
Y
D
L