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Circles
Chapter 9:
(page 328)
Basic Terms
9-1:
(page 329)
O
Q
CIRCLE : the set of all points in a
examples:
at a given distance from a given point.
and
RADIUS : the given
.
in the circle definition.
ALSO any segment joining the
The plural of radius is
CHORD : a segment whose endpoints lie
DIAMETER : a chord that contains the
of the circle to a point on the circle.
.
a circle.
of a circle.
ALSO a diameter has a length equal to
SECANT : a line that contains a
a radius, ie. d =
of a circle.
TANGENT : a line in the plane of a circle that intersects the circle in exactly
The point of intersection is called the point of
.
point.
.
SPHERE : the set of all points in
at a given distance from a given point.
example:
NOTE:
Many of the terms used with circles are also used with spheres.
center:
E
F
radii:
,
,
D
chords:
,
A
secants:
X
C
,
tangent:
B
point of tangency:
n
CONGRUENT CIRCLES : circles that have congruent
.
example:
D
CONGRUENT SPHERES : spheres that have congruent
real life examples:
P
.
CONCENTRIC CIRCLES : circles that lie in the same
same center.
and have the
example:
CONCENTRIC SPHERES : spheres that have the same
.
real life examples:
INSCRIBED POLYGON : a polygon is inscribed in a circle if each
polygon lies on the circle.
of the
CIRCUMSCRIBED CIRCLE : a circle is circumscribed about a polygon if each vertex of
the polygon lies
the circle.
examples:
Assignment: Written Exercises, pages 330 & 331: 1-11 odd #’s
Tangents
9-2:
Theorem 9-1
If a line is tangent to a circle,then the line is
to the radius drawn to the point of tangency.
Given: m is tangent to ΘO at T.
Prove:
(page 333)
O
OT ⊥ m
m
T
Corollary
Z
Tangents to a circle from a point are
.
Given: PA and PB are tangents to ΘO.
A
Prove: PA = PB
P
O
B
Theorem 9-2
If a line in the plane of a circle is
at its outer endpoint, then the line is tangent to the circle.
Given: Line l in the plane of ΘQ;
l ⊥ radius QR at R
Q
Prove: l is tangent to ΘQ.
l
R
to a radius
example:
RS is tangent to ΘP at R. If PR = 5 and RS = 12, find PS.
P
R
S
PS =
CIRCUMSCRIBED POLYGON : a polygon is circumscribed about a circle if each of its
sides is
to the circle.
INSCRIBED CIRCLE : a circle is inscribed in a polygon if each of the sides of the
polygon is
examples:
to the circle.
COMMON TANGENT : a line that is tangent to each of two
circles.
Common internal tangent - intersects the segment joining the
of the circles.
Common external tangent - does not intersect the segment joining the
the circles.
of
TANGENT CIRCLES : two coplanar circles that are tangent to the same ___________
at the same point.
Two tangent circles are either internally tangent ( #
#1
) or externally tangent ( #
#2
Assignment: Written Exercises, page 335: 1-5 odd #’s
).
9-3:
Arcs and Central Angles
(page 339)
A
X
B
Q
Y
CENTRAL ANGLE : an angle with its vertex at the
of a circle.
examples:
ARC : an unbroken part of a
examples:
.
MINOR ARC:
MAJOR ARC:
SEMICIRCLES : if the endpoints of a minor arc are on a
examples:
.
&
MEASURE of a MINOR ARC : equals the measure of its
example:
angle.
! = m!AQB = _______
mAB
The measure of any minor arc is less than
.
MEASURE of a MAJOR ARC : equals 360 minus the measure of its
example:
! = 360º - mAB
" = 360º - ______ = ______
mAYB
The measure of any major arc is between
MEASURE of a SEMICIRCLE : equals
example:
arc.
and
.
! = m!XBY
! = ________
m XAY
.
ADJACENT ARCS : (of a circle) are arcs with exactly
example:
point in common.
&
Y
X
example:
&
Z
W
Arc Addition Postulate
Postulate 16
The measure of the arc formed by two
the sum of the measures of these two arcs.
example:
arcs is
In ΘE, find the measure of the angle or the arc named.
A
(1) m!1 = _______
! = _______
(2) mBC
! = _______
(3) mAC
" = _______
(4) mADB
E
1
B
80º
D
80º
C
CONGRUENT ARCS : arcs in the same circle or congruent circles that have
measures.
example: ____________ ≅ ____________
Theorem 9-3
In the same circle or in congruent circles, two minor arcs are congruent,
if and only if their central angles are congruent.
A
If !1 " !2, then _______ " _______
! , ! CD
! then _______ ! _______.
If AB
D
1
2
B
C
Assignment: Written Exercises, pages 341 & 342: 1 - 8 ALL #’s, 17 – 20 ALL #’s
Arcs and Chords
9-4:
(page 344)
Y
The minor arc,
, is the arc of chord
.
X
Theorem 9-4
Z
In the same circle or in congruent circles:
(1) Congruent arcs have congruent
.
(2) Congruent chords have congruent
(1)
T
R
.
(2)
T
R
O
U
Q
S
" ! TU
"
Given: ! O; RS
Prove: RS ! TU
U
S
Given: ! Q ; RS ! TU
" ! TU
"
Prove: RS
! if XY
" ! YZ
".
A point Y is called the ______________ of XYZ
X
Z
Y
Theorem 9-5
A diameter that is perpendicular to a chord
and its arc.
Given: ! O; CD ! AB
" " BD
"
Prove: AZ " BZ ; AD
the chord
C
O
A
Z
D
B
Theorem 9-6
In the same circle or in congruent circles:
(1) Chords equally distant from the center(s) are
.
(2) Congruent chords are equally
(1)
from the center(s).
(2)
A
A
X
X
B
B
O
D
Q
Y
D
C
Given: ! O ; OX = OY
Y
C
Given: ! Q ; AB ! CD
Prove: QX = QY
Prove: AB ! CD
examples:
(1)
If PS = 12 & TR = 15,
then find QR.
60º
S
(2)
R
70º
T
A
Q
50º
P
T
S
Q
R
160º
QR = __________
(3)
In ΘA, SQ = 12 & AT = 8,
then find PR.
P
PR = __________
G
In ΘO, FL = 3, GO = 5, & OP = 4, then find HJ.
L
HJ = __________
O
F
H
P
Assignment: Written Exercises, page 347: 1-9 odd #’s
J
Inscribed Angles
9-5:
INSCRIBED ANGLE: an angle whose vertex is
contain chords of the circle.
(page 349)
a circle and whose
A
example:
intercepts
B
Theorem 9-7
C
The measure of an inscribed angle is equal to
measure of its intercepted arc.
the
Given: !ABC inscribed in ! O
1 "
Prove: m!ABC = mAC
2
B
B
O
A
B
O
A
C
A
C
O
C
Case I:
Pt. O lies on ∠ABC
Example 1:
Case II:
Pt. O lies inside ∠ ABC
Case III:
Pt. O lies outside ∠ABC
Find the values of “x”, “y”, and “z”.
x=
90º
xº
yº
y=
z=
80º
55º
zº
Corollary 1
If two inscribed angles intercept the same arc, then the angles are
Corollary 2
An angle inscribed in a semicircle is a
Corollary 3
If a quadrilateral is inscribed in a circle, then its opposite angles are
Example 2:
Find the measure of each numbered angle.
.
angle.
.
120º
(a)
(b)
(c)
140º
3
5
70º
2
1
4
90º
m∠1 =
; m ∠2 =
;
m∠3 =
;
m ∠4 =
; m∠ 5 =
Theorem 9-8
The measure of an angle formed by a chord and a tangent is equal to half
the measure of the
arc.
Given: PT is tangent to ! O
1 "
Prove: m!ATP = mAT
2
A
O
P
T
Example 3:
" = 260º , find m XY
# and m!PXY.
If XP is tangent to ! A and m XZY
Z
X
A
P
Y
! = __________
m XY
m!PXY = __________
Assignment: Written Exercises, page 355: 19
Worksheet on Lesson 9-5: Inscribed Angles
Other Angles
9-6:
(page 357)
The measure of an angle formed by two chords that intersect inside a
circle is equal to half the sum of the measures of the intercepted arcs.
Theorem 9-9
A
D
Given: Chords AB and CD intersect inside a circle.
1 !
!)
Prove: m!1 = (mAC
+ mBD
2
1
B
C
examples:
! = 45º and mRS
! = 75º, then m!1 = _______.
(1) If mPQ
R
Q
1
P
S
! = 80º, then mPQ
! = _______.
(2) If m!1 = 55º and mRS
R
Q
1
P
S
Theorem 9-10
Case I: 2
The measure of an angle formed by two secants, two tangents, or
a secant and a tangent drawn from a point outside a circle is
equal to half the difference of the measures of the intercepted arcs.
secants
m!1 =
Case II: 2
1
(x - y)
2
xº
yº 1
tangents
1
m!2 = (x - y)
2
Case III: secant
xº
yº
2
& tangent
1
m!3 = (x - y)
2
yº
xº
3
examples:
! = 100º and mEB
! = 40º, then m!A = _______.
(3) If mDC
D
E
A
C
B
! = 70º, then m XVY
" = _______.
(4) If m!W = 65º and m XZ
Y
Z
W
V
X
! = 240º, then mQS
" = _______ and m!P = _______.
(5) If mQRS
S
R
P
Q
Assignment: Written Exercises, page 359: 1-10 ALL #’s
Worksheet on Lesson 9-6: Other Angles
Circles and Lengths of Segments
9-7:
(page 361)
S
PQ and RS are
.
Q
X
PX and XQ are segments of
PQ .
R
RX and XS are segments of chord
Theorem 9-11
.
P
When two chords intersect inside a circle, the
of the
segments of one chord equals the product of the segments of the other
chord.
D
Given: AB and CD intersect at P.
Prove:
A
u
r
r· s = t · u
P
C
t
s
B
Example 1:
(a)
If AP = 4, PB = 6, and CP = 8,
then PD = __________.
If AP = 4, PB = 9, and CD is bisected,
then CP = _________.
D
A
D
A
P
(b)
B
P
C
C
B
DF and HF are
segments.
EF and GF are
segments of those secants.
D
E
F
G
Theorem 9-12
H
When two secant segments are drawn to a circle from an external point,
the product of one secant segment and its external segment equals
the product of the other secant segment and its external segment.
Given: PA and PC drawn to the circle from point P
Prove:
r· s = t · u
A
r
B
s
P
u
D
C
t
Example 2:
(a)
If PQ = 9, QR = 3, and TR = 4,
then SR = __________.
P
S
Q
T
R
(b)
If PR = 15, SR = 12, and TR = 5,
then QR = __________.
P
S
Q
T
R
PA is a
segment.
PC is a
segment.
C
B
P
A
Theorem 9-13
When a secant segment and a tangent segment are drawn to a circle from
an external point, the product of the secant segment and its external
segment is equal to the square of the tangent segment.
Given: Secant segment PA and tangent segment PC drawn to the circle from P.
Prove:
r · s = t2
|-------------------- r ----------------------|
B
A
s
P
t
C
Example 3:
(a)
If RT = 25 and RS = 5,
then QR = __________.
(b)
If QR = 6 and RT = 9,
then RS = __________.
T
Q
T
Q
S
R
S
R
Assignment: Written Exercises, page 366: 23
Worksheet on Lesson 9-7: Circles and Lengths of Segments
Prepare for Test on Chapter 9: Circles