Chapter 10: Circles

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Chapter 10 Circles
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10.1 Circles and Circumference

• Name a circle by the letter at the center of the
  circle
• Diameter- segment that extends from one point on
  the circle to another point on the circle through
  the center point
• Radius- segment that extends from one point on
  the circle to the center point
• Chord- segment that extends from one point on the
  circle to another point on the circle
• Diameter2 x radius (d2r)
• Circumference the distance around the circle
• C2pr or C pd

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Circle X Diameter- Radius- Chord-
chord
A
B
diameter
E
C
X
radius
D
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1. Name the circle
2. Name the radii
3. Identify a chord
4. Identify a diameter

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1. Find the circumference of a circle to the nearest
   hundredth if its radius is 5.3 meters.

b. Find the diameter and the radius of a circle
to the nearest hundredth if the circumference of
the circle is 65.4 feet.
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10.2 Angles, Arcs and Chords

• 10.2
• Semi-circle half the circle (180 degrees)
• Minor arc less than 180 degrees
• Name with two letters
• Major arc more than 180 degrees
• Name with three letters
• Minor arc central angle
• Arc length

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B
Minor arc
Minor arc
Minor arc AB or BC Semicircle ABC or
CDA Major arc ABD or CBD AB BC 180
C
Central angle
X
A
D
Semicircle
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Find the value of x.
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• Find x and angle AZE

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Find the measure of each minor arc.
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10.3 Arcs and Chords

• If two chords are congruent, then their arcs are
  also congruent
• In inscribed quadrilaterals, the opposite angles
  are supplementary
• If a radius or diameter is perpendicular to a
  chord, it bisects the chord and its arc
• If two chords are equidistant from the center of
  the circle, the chords are congruent

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A
If FEBC, then arc FE arc BC Quad. BCEF is an
inscribed polygon opposite angles are
supplementary angles B E 180 angles F C
180 Diameter AD is perpendicular to chord EC
so chord EC and arc EC are bisected
B
F
C
E
D
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B
You will need to draw in the radius yourself
E
You can use the pythagorean theorem to find the
radius when a chord is perpendicular to a
segment from the center XE XF so chord AB
chord CD because they are equidistant from the
center
A
X
C
F
D
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10.4 Inscribed Angles

• Inscribed angle an angle inside the circle with
  sides that are chords and a vertex on the edge of
  the circle
• Inscribed angle ½ intercepted arc
• An inscribed right angle, always intercepts a
  semicircle
• If two or more inscribed angles intercept the
  same arc, they are congruent

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A
Inscribed angles angle BAC, angle CAD, angle
DAE, angle BAD, angle BAE, angle CAE
X
B
Ex Angle DAE ½ arc DE
C
E
D
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A
Inscribed angle BAC intercepts a semicircle- so
angle BAC 90 Inscribed angles GDF and GEF both
intercept arc GF, so the angles are congruent
B
C
D
F
E
G
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A. Find m?X.
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Refer to the figure. Find the measure of angles
1, 2, 3 and 4.
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ALGEBRA Find m?R.
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ALGEBRA Find m?I.
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ALGEBRA Find m?B.
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ALGEBRA Find m?D.
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The insignia shown is a quadrilateral inscribed
in a circle. Find m?S and m?T.
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10.5 Tangents

• Tangent a line that shares only one point with a
  circle and is perpendicular to the radius or
  diameter at that point.
• Point of tangency the point that a tangent
  shares with a circle
• Two lines that are tangent to the same circle and
  meet at a point, are congruent from that point to
  the points of tangency

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Lines AC and AF are tangent to circle X at points
B and E respectively -B and E are points of
tangency Radius XB is perpendicular to tangent
AC at the point of tangency AE and AB are
congruent because they are tangent to the same
circle from the same point
X
C
F
E
B
A
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A. Copy the figure and draw the common tangents
to determine how many there are. If no common
tangent exists, choose no common tangent.
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10.6 Secants, Tangents, and Angle Measures

• Secant and Tangent
• Interior angle ½ intercepted arc
• Two Secants
• Interior angle ½ (sum of intercepted arcs)
• Two Secants
• Exterior angle ½ (far arc close arc)
• Two Tangents
• Exterior angle ½ (far arc close arc)

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C
2 Secants/chords Angle 1 ½ (arc AD arc
CB) Angle 2 ½ (arc AC arc DB)
B
2
1
A
D
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E
Secant ED intersects tangent FC at a point of
tangency (point F) Angle 1 ½ arc FE Angle 2 ½
(arc EA arc FB)
1
F
B
A
2
D
C
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A. Find x.
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B. Find x.
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C. Find x.
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A. Find m?QPS.
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10.7 Special Segments in a Circle

• Two Chords
• seg1 x seg2 seg1 x seg2
• Two Secants
• outer segment x whole secant
• outer segment x whole secant
• Secant and Tangent
• outer segment x whole secant tangent squared

Add the segments to get the whole secant
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E
D
A
F
2 chords AO x OB DO x OC 2 secants EF x EG
EH x EI
H
O
C
I
B
G
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A
D
Secant and Tangent AD x AB AC x AC
C
B
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A. Find x.
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B. Find x.
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A. Find x.
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B. Find x.
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Find x.
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Find x.
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Find x. Assume that segments that appear to be
tangent are tangent.