Transcript Interactive Chalkboard

10.3 Arcs and Chords
Glencoe Geometry Interactive Chalkboard
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Glencoe/McGraw-Hill
8787 Orion Place
Columbus, Ohio 43240
Objectives
• Recognize and use relationships between
arcs and chords
• Recognize and use relationships between
chords and diameters
Arcs and Chords S
• The endpoints of chords
also create arcs.
R
A
B
Theorem 10.2: In a or  s, two minor
arcs are  iff their corresponding chords
are .
PROOF Write a proof.
Given:
is a semicircle.
Prove:
Proof:
Statements
Reasons
1.
1. Given
is a semicircle.
2.
2. Def. of semicircle
3.
3. In a circle, 2 chords
are , corr. minor
arcs are .
4.
4. Def. of
5.
5. Def. of arc measure
arcs
Answer:
Statements
Reasons
6.
6. Arc Addition Postulate
7.
7. Substitution
8.
8. Subtraction Property
and simplify
9.
9. Division Property
10.
10. Def. of arc measure
11.
11. Substitution
PROOF Write a proof.
Given:
Prove:
Proof:
Statements
Reasons
1.
1. Given
2.
2. In a circle, 2 minor
arcs are , chords
are .
3.
3. Transitive Property
4.
4. In a circle, 2 chords
are , minor arcs
are .
Arcs and Chords
• The chords of adjacent arcs
can form a polygon.
• The polygon in the picture is
an inscribed polygon
because all of the vertices lie
on the circle and the circle is
a circumscribed circle
because it contains all of the
vertices of the polygon.
TESSELLATIONS The rotations of a tessellation can
create twelve congruent central angles. Determine
whether
.
Because all of the twelve central angles are congruent,
the measure of each angle is
Let the center of the circle be A. The measure of
Then
.
The measure of
Then
.
Answer: Since the measures of
equal,
.
are
ADVERTISING A logo for an advertising campaign is a
pentagon that has five congruent central angles.
Determine whether
.
Answer: no
Diameters and Chords
• Diameters that are ┴ to a chord in a
create special segments and arc
relationships.
• Theorem 10.3: In a , if a diameter (or
radius) is ┴ to a chord, then it bisects the
chord and its arc.
Diameters and Chords
Theorem 10.3 (continued):
If JK ┴ LM, then MO  LO
and arc LK  arc MK.
o
Circle W has a radius of 10 centimeters. Radius
is
perpendicular to chord
which is 16 centimeters
long.
If
find
Since radius
is perpendicular to chord
Arc addition postulate
Substitution
Substitution
Subtract 53 from each side.
Answer: 127
Circle W has a radius of 10 centimeters. Radius
is
perpendicular to chord
which is 16 centimeters
long.
Find JL.
Draw radius

A radius perpendicular to a
chord bisects it.
Definition of segment
bisector
Use the Pythagorean Theorem to find WJ.
Pythagorean Theorem
Simplify.
Subtract 64 from each side.
Take the square root of
each side.
Segment addition
Subtract 6 from each side.
Answer: 4
Circle O has a radius of 25 units. Radius
is
perpendicular to chord
which is 40 units long.
a. If
Answer: 145
b. Find CH.
Answer: 10
More About Chords
• Theorem 10.4: In a
or in 
s, two
chords are  iff they are equidistant from
the center.
Chords
and
If the radius of
are equidistant from the center.
is 15 and EF = 24, find PR and RH.
are equidistant from P, so
.
Draw
to form a right triangle. Use the Pythagorean
Theorem.
Pythagorean Theorem
Simplify.
Subtract 144 from each side.
Take the square root of
each side.
Answer:
Chords
and
are equidistant from the center of
If TX is 39 and XY is 15, find WZ and UV.
Answer:
Assignment
• Pre-AP Geometry
Pg. 540 #11 – 36, 40, 42
• Geometry:
Pg. 540 #11 – 35