Transcript Lesson 6.1 Chord Properties (2014)

Geometry
Lesson 6.1
Chord Properties
Geometry
Angles in a Circle
• In a plane, an angle
whose vertex is the
center of a circle is
a central angle of
the circle.
• In a plane, an angle
whose vertex is on
the circle is an
inscribed angle of
the circle.
inscribed angle
central angle
A
major
arc
minor
arc
P
Q
B
C
Arc Measures
Geometry
G
The measure of a
minor arc is defined
to be the measure of
its central angle.
60°
60°
JE
H
F
E
180°
Geometry
Chord Central Angle Conjecture
• If two chords in a circle
are , then their central
angles are .
A
X
•AXB  CXB
if and only if
AB  BC
C
B
Geometry
Chord Arcs Conjecture
• If two chords in a
circle are , then
their intercepted
arcs are .
A

• AB  BC if and
only if AB  BC
X
C
B
Geometry
Chord Distance to
Center Conjecture
• Two  chords in a
circle are
equidistant from the
center of the circle.
Geometry
Perpendicular to a Chord
Conjecture
• The perpendicular
from the center of a
circle to a chord is
the bisector of the
chord.
J
M
JK is a diameter of
the circle.
K
L
Geometry
Perpendicular Bisector
of a Chord Conjecture
• The perpendicular
bisector of a chord
passes through the
center of a circle.
J
M
K
L
Geometry
Summary of
Chord Property #2, #3 and #4
• A perpendicular line
from the center of a
chord to the center
of a circle:
#2: Makes a 90° angle
with the chord
#3: Creates two equal
line segments RS
and QR
#4: Must pass through
the center of the
circle O
Ex. 1
Geometry
(x + 40)°
• You can use
Theorem 10.4 to
find m AD .



 
C
A
• Because AD  DC,
and AD  DC . So,
m AD = m DC
2x = x + 40
x = 40
2x°
B
Substitute
Subtract x from each
side.
Geometry
Ex. 2
AB = 8; DE = 8, and
CD = 5. Find CF.
A
8 F
B
C
E
5
8
G
D
Geometry
Let’s get crazy…
•Find b.
3
8
b
12
Geometry
Let’s get crazy…
•Find b.
3
8
b
12
•Step 1: What do we need to find?
•We need a radius to complete this big triangle.
Geometry
Let’s get crazy…
•Find b.
3
8
b
12
•How do we find a radius?
•We can draw multiple radii (radiuses).
Geometry
Let’s get crazy…
•Find b.
3
8
b
12
•How do we find a radius?
•Now what do we have and what will we do?
Geometry
Let’s get crazy…
•Find b.
3
8
b
12
•How do we find a radius?
•We create this triangle but how do we get the missing side?
Geometry
Let’s get crazy…
•Find b.
3 4
8
b
12
•How do we find a radius?
•We create this triangle but how do we get the missing side?
Geometry
Let’s get crazy…
•Find b.
5
3 4
13
b
•5
12
•!!!Pythagorean Theorem!!!
•a2+b2=c2 so 52+122=169 so b is 13.