CIRCLES Everything you wanted to know and then some!!
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CIRCLESEverything you wanted to know and then some!!
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Parts of a CircleCircle – set of all points _________ from a given point called the _____ of the circle.
C
Symbol:
equidistant
center
C
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Secant Line
A secant line intersects the circle at exactly TWO points.
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TANGENT: a LINE that intersects the circle exactly ONE
time
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Name the term that best describes the line.
Secant
Radius
DiameterChordTangent
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If a line (segment or ray) is tangent to a circle, then it is perpendicular to the radius drawn to the point of tangency.
Point of Tangenc
y
More Pythagorean Theorem type problems! Yeah!!
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a2 + b2 = c2
x = 1592 + 122 = x2
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R
S
T
TSRS If two segments from
the same exterior point are tangent to a circle, then they are
congruent.
Party hat problems!
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A
C
B152 x
x14
15x
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P
A
BC
Central Angle : An Angle whose vertex is at the center of the
circleMinor ArcMajor Arc
Less than 180°
More than 180°
ABACBTo name:
use 2 letters
To name: use 3 letters
APB is a Central Angle
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measure of an arc = measure of central angleA
B
C
Q 96
m AB
m ACBm AE
E
=
==
96°
264°84°
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Tell me the measure of the following arcs.
80100
40
140A
B
C
DR
m DAB =m BCA =
240
260
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Inscribed Angle: An angle whose vertex is on the circle and whose sides are chords of the circle
INSCRIBEDANGLE
INTERCEPTED
ARC
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2 ArcdIntercepteAngleInscribed
160
80
To find the measure of an inscribed angle…
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120
x
What do we call this type of angle?What is the value of x?
y
What do we call this type of angle?How do we solve for y?The measure of the inscribed angle is HALF the measure of the
inscribed arc!!
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Examples3. If m JK = 80, find m JMK.
M
Q
K
S
J
4. If m MKS = 56, find m MS.
40
112
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Case I: Vertex is ON the circle
ANGLE = ARC2
ANGLE
ARCANGLE
ARC
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Ex. 2 Find m1.
84°
1
m1 = 42
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Case II: Vertex is inside the circleA
B
CD
ANGLE = (ARC + ARC)2
ANGLE
ARC
ARC
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Ex. 4 Find m1. AB
CD
1
93°
113°
m1 = 103
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Case III: Vertex is outside the circleA
B
C
D
ANGLE = (Large ARC Small ARC)2
ANGLE
LARGE ARC
small
ARC
ANGLELARGE
ARC
small ARC sm
all ARCLARGE
ARC
ANGL
E
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A
B
Ex. 7 Find mAB.
27°
70°
mAB = 16
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1
Ex. 8 Find m1.26
0°
m1 = 80
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a
bc
d ab = cd
part part = part part
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9
2
6xx = 3
32
12
x
x = 83
26
x
x = 1
Ex: 1 Solve for x.
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E A B
C
D
EA • EB = EC • EDoutside whole = outside whole
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EA
B
C
D
7 13
4
x
7(7 + 13) 4(4 + x)=
Ex: 3 Solve for x.
140 = 16 + 4x124 = 4x
x = 31
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E
A
B C
EA2 = EB • ECoutside whole = outside whole
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E
A
B C
24
12 x
outside whole = outside whole 242 = 12 (12 + x)576 = 144 + 12x x = 36
Ex: 5 Solve for x.