Transcript Chapter 10: Circles

Chapter 10: Circles
10.1 Circles and
Circumference
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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
Diameter=2 x radius (d=2r)
Circumference: the distance around the circle
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C=2πr or
C= πd
Circle X
A
chord
Diameter- EC
B
diameter
E
X
radius
D
C
Radius-
DX
Chord-
AB
1. Name the circle
2. Name the radii
3. Identify a chord
4. Identify a diameter
a.
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.
10.2 Angles, Arcs and Chords
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10.2
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Semi-circle: half the circle (180 degrees)
Minor arc: less than 180 degrees
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Major arc: more than 180 degrees
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Name with two letters
Name with three letters
Minor arc = central angle
Arc length: arc  arclength
360
2r
B
Minor arc
Minor arc
C
Minor arc = AB
or BC
Semicircle =
ABC or CDA
Central
angle
X
Major arc = ABD
or CBD
A
AB + BC = 180
D
Semicircle
Find the value of x.
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Find x and angle AZE
10.3 Arcs and Chords
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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
A
If FE=BC, then arc FE =
arc BC
B
Quad. BCEF is an
inscribed polygon –
opposite angles are
supplementary angles B
and E & angles F and C
F
C
E
D
Diameter AD is
perpendicular to chord EC
– so chord EC and arc EC
are bisected
B
*You can use the pythagorean
theorem to find the radius
when a chord is perpendicular
to a segment from the center
E
A
X
C
F
EX = FX so chords AB and CD
are congruent because they are
equidistant from the center
D
In the circle below, diameter QS is 14 inches
long and chord RT is 10 inches long. Find
VU.
10.4 Inscribed Angles
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Inscribed angle: an angle inside the circle
with sides that are chords and a vertex on
the edge of the circle
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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
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
A
Inscribed angle BAC
intercepts a semicircle- so
angle BAC =90
B
C
D
F
E
G
Inscribed angles GDF and
GEF both intercept arc
GF, so the angles are
congruent
A. Find mX.
Refer to the figure. Find the measure of
angles 1, 2, 3 and 4.
ALGEBRA Find mR.
ALGEBRA Find mI.
ALGEBRA Find mB.
ALGEBRA Find mD.
INSIGNIAS An insignia is an emblem that
signifies rank, achievement, membership, and so
on. The insignia shown is a quadrilateral
inscribed in a circle. Find mS and mT.
10.5 Tangents
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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
Lines AC and AF are
tangent to circle X at points
B and E respectively
-B and E are
points of tangency
C
F
E
B
A
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
A. Copy the figure and draw the common
tangents to determine how many there are.
If no common tangent exists, choose no
common tangent.
10.6 Secants, Tangents, and
Angle Measures
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Secant and Tangent
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Two Secants:
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Interior angle = ½ (sum of intercepted arcs)
Two Secants
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Interior angle = ½ intercepted arc
Exterior angle = ½ (far arc – close arc)
Two Tangents
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Exterior angle = ½ (far arc – close arc)
C
2 Secants/chords:
B
2
A
D
1
Angle 1 = ½ (arc AD + arc CB)
Angle 2 = ½ (arc AC + arc DB)
E
Secant ED intersects tangent FC
at a point of tangency (point F)
Angle 1 = ½ arc FE
Angle 2 = arc EA – arc FB
1
F
2
B
A
D
C
A. Find x.
B. Find x.
C. Find x.
A. Find mQPS.
B.
A. Find mFGI.
A.
B.
10.7 Special Segments in a
Circle
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Two Chords
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Two Secants
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seg1 x seg2 = seg1 x seg2
outer segment x whole secant =
outer segment x whole secant
Secant and Tangent
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outer segment x whole secant = tangent squared
*Add the segments to get the whole secant
E
D
A
F
2 chords:
H
AO x OB = DO x OC
2 secants:
O
EF x EG = EH x EI
C
I
B
G
A
D
Secant and Tangent:
AD x AB = AC x AC
C
B
A. Find x.
B. Find x.
A. Find x.
B. Find x.
Find x. Round to the
nearest tenth.
Find x.
Find x.
LM is tangent to the circle. Find x. Round to the
nearest tenth.
Find x. Assume that segments that appear
to be tangent are tangent.