Transcript 10.2 Arcs and Chords

Geometry
Chord Lengths
Section 6.3
Geometry
Mrs. Spitz
Spring 2005
Modified By Mr. Moss, Spring
2011
Geometry
Today’s Standards
• MM2G3. Students will understand
the properties of circles.
• a. Understand and use properties of
chords, tangents, and secants as an
application of triangle similarity.
• d. Justify measurements and
relationships in circles using geometric
and algebraic properties.
Geometry
Using Chords of Circles
• A point Y is called
the midpoint of
if XY  YZ . Any
line, segment, or ray
that contains Y
bisects X YZ .
 

Geometry
Theorem 6.5 – covered already
• In the same circle, or in
congruent circles, two
minor arcs are
congruent if and only if
their corresponding
chords are congruent.

AB  BC if and only if
AB  BC
A
C
B
Ex. 4: Using Theorem 6.5
Geometry
D
• You can use
Theorem 6.5 to find
m AD



 
2x°
C
A
• Because AD  DC,
and AD  DC . So,
m AD = m DC
2x = x + 40
x = 40

(x + 40)°
B
Substitute
Subtract x from each side.
AD = 2(40) = 80° Substitute
Geometry
Theorem 6.6
• If one chord is a
perpendicular
bisector of another
chord, then the first
chord is a diameter.
J
M
JK is a diameter of
the circle.
K
L
Geometry
Ex. 5: Finding the Center of a
Circle
• Theorem 6.6 can be
used to locate a
circle’s center as
shown in the next
few slides.
• Step 1: Draw any
two chords that are
not parallel to each
other.
Geometry
Ex. 5: Finding the Center of a
Circle
• Step 2: Draw the
perpendicular
bisector of each
chord. These are
the diameters.
Geometry
Ex. 5: Finding the Center of a
Circle
• Step 3: The
perpendicular
bisectors intersect
at the circle’s
center.
center
Geometry
Theorem 6.7
• If a diameter of a
circle is
perpendicular to a
chord, then the
diameter bisects the
chord and its arc.
F
E
 
G
DE  EF ,
DG  GF
D
Geometry
Theorem 6.7 Proof
If diameter is  to a Chord
Then BE  DE
•
•
•
•
Construct radii PB & PD
A
Diameter AC  BD - given
PE = PE – reflexive prop
PD = PB – both radii of
same 
• PBE  PDE by HL
• BE  DE by CPCTC
B
P
E
D
C
Geometry
Theorem 6.8
• In the same circle,
or in congruent
circles, two chords
are congruent if and
only if they are
equidistant from the
center.
• AB  AC if and only
if DE  GF.
D
E
B
F
A
C
G
Geometry
Theorem 6.8 Proof (forwards):
If 2 chords are 
D
Then AB = AC
E
•
•
•
•
•
•
•
•
Draw radii AE and AF
DE  FG – given
B and C are midpoints
BE CF
EBA = FCA = 90°
AE  AF – both radii
ABE  ACF by HL
AB = AC - CPCTC
B
F
A
C
G
Geometry
Theorem 6.8 Proof (reverse):
If AB = AC
Then 2 chords are 
•
•
•
•
•
•
•
•
Draw radii AE and AF
AB = AC – given
EBA = FCA = 90°
AE  AF – both radii
ABE  ACF by HL
BE = CF – CPCTC
BD = CG by similar
DE = FG by addition
D
E
B
F
A
C
G
Geometry
Practice
• 11-3 Study Guide
• 11-4 Practice – skip problems 1 & 2
• HW: Pg 201, # 1 – 13 all
Geometry
Ex. 7: Using Theorem 6.8
AB = 8; DE = 8, and
CD = 5. Find CF.
A
8 F
B
C
E
5
8
G
D