definitions a circle is the set of all points in a plane that are equidistant from a given point...
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Definitions
A circle is the set of all points in a plane that are equidistant from a given point called the center of the circle.
Radius – the distance from the center to a point on the circle
Congruent circles – circles that have the same radius.Diameter – the distance across the circle through its
center
Definition
Tangent – a line in the plane of a circle that intersects the circle in exactly one point.
ST is a tangent
S
T
Example 1
Tell whether the line or segment is best described as a chord, a secant, a tangent, a diameter, or a radius.
FC
B
G
A
H
D
E
Id. CE
c. DF
b. EI
a. AH tangent
diameter
chord
radius
More definitions
Interior of a circle – consists of the points that are inside the circle
Exterior of a circle – consists of the points that are outside the circle
Definition
Point of tangency – the point at which a tangent line intersects the circle to which it is tangent
point of tangency
Perpendicular Tangent Theorem
If a line is tangent to a circle, then it is perpendicular to the radius drawn to the point of tangency.
l
Q
P
If l is tangent to Q at P, then l QP.
Definitions
Minor arc – Part of a circle that measures less than 180°
Major arc – Part of a circle that measures between 180° and 360°.
Semicircle – An arc whose endpoints are the endpoints of a diameter of the circle.
Note : major arcs and semicircles are named with three points and minor arcs are named with two points
Definitions
Measure of a minor arc – the measure of its central angle
Measure of a major arc – the difference between 360° and the measure of its associated minor arc.
Arcs and Chords Theorem
In the same circle, or in congruent circles, two minor arcs are congruent if and only if their corresponding chords are congruent.
A
B
C
AB BC if and only if AB BC
Perpendicular Diameter Theorem
If a diameter of a circle is perpendicular to a chord, then the diameter bisects the chord and its arc.
D
F
G
EDE EF, DG FG
Right Triangles Pythagorean Theorem
45
43
11
D
E
C
Radius is perpendicular to the tangent. < E is a right angle
Example
AB is tangent to C at B.AD is tangent to C at D.
Find the value of x.
11
x2 + 2
AC
D
BAD = AB
x2 + 2 = 11
x2 = 9
x = 3
Example
Find the measures of the red arcs. Are the arcs congruent?
41
41
AC
D
EmAC = mDE = 41Since the arcs are in the same circle, they are congruent!
Example 3
Find the measures of the red arcs. Are the arcs congruent?
81
C
A
D
E
mDE = mAC = 81However, since the arcs are not of the same circle orcongruent circles, they are NOT congruent!
Definitions
Inscribed angle – an angle whose vertex is on a circle and whose sides contain chords of the circle
Intercepted arc – the arc that lies in the interior of an inscribed angle and has endpoints on the angle
inscribed angle
intercepted arc
Measure of an Inscribed Angle Theorem
If an angle is inscribed in a circle, then its measure is half the measure of its intercepted arc.
C
A
D BmADB =
1
2mAB
Example 1
Find the measure of the blue arc or angle.
RS
QT
a.
mQTS = 2(90 ) = 180
b.80
E
FG
mEFG = 1
2(80 ) = 40
Congruent Inscribed Angles TheoremIf two inscribed angles of a circle intercept
the same arc, then the angles are congruent.
A
CB
D
C D
Example 2
It is given that mE = 75 . What is mF?
D
E
HF
Since E and F both interceptthe same arc, we know that theangles must be congruent.
mF = 75
Definitions
Inscribed polygon – a polygon whose vertices all lie on a circle.
Circumscribed circle – A circle with an inscribed polygon.
The polygon is an inscribed polygon and the circle is a circumscribed circle.
Inscribed Right Triangle Theorem
If a right triangle is inscribed in a circle, then the hypotenuse is a diameter of the circle. Conversely, if one side of an inscribed triangle is a diameter of the circle, then the triangle is a right triangle and the angle opposite the diameter is the right angle.
A
C
BB is a right angle if and only if ACis a diameter of the circle.
Inscribed Quadrilateral Theorem
A quadrilateral can be inscribed in a circle if and only if its opposite angles are supplementary.
C
E
F
DG
D, E, F, and G lie on some circle, C if and only if mD + mF = 180 and mE + mG = 180 .
Example 3
Find the value of each variable.
2x
Q
A
B
C
a.
2x = 90
x = 45
b. z
y
80
120
D
E
F
G
mD + mF = 180
z + 80 = 180
z = 100
mG + mE = 180
y + 120 = 180
y = 60
Tangent-Chord Theorem
If a tangent and a chord intersect at a point on a circle, then the measure of each angle formed is one half the measure of its intercepted arc.
21
B
A
Cm1 = 1
2mAB
m2 = 1
2mBCA
Example 2
(9x+20)
5x
D
B
CA
BC is tangent to the circle. Find mCBD.
2(5x) = 9x + 20
10x = 9x + 20
x = 20
mCBD = 5(20 )
mCBD = 100
Interior Intersection Theorem
If two chords intersect in the interior of a circle, then the measure of each angle is one half the sum of the measures of the arcs intercepted by the angle and its vertical angle.
m1 = 1
2(mCD + mAB)
m2 = 1
2(mAD + mBC)
21
A
C
D
B
Exterior Intersection Theorem
If a tangent and a secant, two tangents, or two secants intersect in the exterior of a circle, then the measure of the angle formed is one half the difference of the measures of the intercepted arcs.
Diagrams for Exterior Intersection Theorem
1
BA
C
m1 = 1
2(mBC - mAC)
2
P
RQ
m2 = 1
2(mPQR - mPR)
3
X
W
YZ
m3 = 1
2(mXY - mWZ)
Example 3
Find the value of x.
174
106
x
P
R
Q
S
x = 1
2(mPS + mRQ)
x = 1
2(106+174 )
x = 1
2(280)
x = 140
Try This!
Find the value of x.
120
40
x
T
R
S
U
x = 1
2(mST + mRU)
x = 1
2(40+120 )
x = 1
2(160)
x = 80
Example 5
Find the value of x.
mABC = 360 - 92
mABC = 268 x92
C
AB
x = 1
2(268 - 92)
x = 1
2(176)
x = 88
Chord Product Theorem
If two chords intersect in the interior of a circle, then the product of the lengths of the segments of one chord is equal to the product of the lengths of the segments of the other chord.
E
C
D
A
B
EA EB = EC ED
Secant-Secant Theorem
If two secant segments share the same endpoint outside a circle, then the product of the length of one secant segment and the length of its external segment equals the product of the length of the other secant segment and the length of its external segment.
C
A
B
ED
EA EB = EC ED
Secant-Tangent Theorem
If a secant segment and a tangent segment share an endpoint outside a circle, then the product of the length of the secant segment and the length of its external segment equals the square of the length of the tangent segment.
C
A
E
D
(EA)2 = EC ED
Example 2
Find the value of x.
LM LN = LO LP
9(20) = 10(10+x)
180 = 100 + 10x
80 = 10x
x = 8 x
10
11
9
O
M
N
L
P
Try This!
Find the value of x.
x
1012
11
H
GF
E
D
DE DF = DG DH
11(21) = 12(12 + x)
231 = 144 + 12x
87 = 12x
x = 7.25
Example 3
Find the value of x.
x
12
24
D
BC
A
CB2 = CD(CA)
242 = 12(12 + x)
576 = 144 + 12x
432 = 12x
x = 36