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Circles
Notes 10-1 p3
Objectives: Identify and use parts of circles
Solve Problems involving the circumference of a circle.
A CIRCLE is the LOCUS of all points in a plane
EQUIDISTANT from a given point, which is the CENTER of
the circle.
A circle is usually named by its CENTER point.
Any SEGMENT with ENDPOINTS on the circle is a
CHORD of the circle.
A CHORD that contains the CENTER of the circle is a
DIAMETER of the circle.
Any SEGMENT with ENDPOINTS that are the CENTER and
a POINT on the CIRCLE is a RADIUS.
All RADII of a circle are CONGRUENT and all DIAMETERS
are CONGRUENT.
Circles
Notes 10-1 p4
Name the circle.
Name a radius of the circle.
Name a chord of the circle.
Name a diameter of the circle.
If ST = 18, find RS.
If RM = 24, find QM.
If RN = 2, find RP.
Circles
Notes 10-1 p5
The diameter of
The diameter of
The diameter of
is 22 mm.
is 16 mm.
is 10 mm.
Find EZ.
Find XF.
Find YF.
The CIRCUMFERENCE of a circle is the DISTANCE AROUND the
circle.
The RATIO of the CIRCUMFERENCE to the DIAMETER of a
"
CIRCUMFERENCE equals "
circle is always equal to PI
.
times DIAMETER
CIRCUMFERENCE equals TWO times
"
OR
times RADIUS.
For a circumference of C units and a diameter of d units or a
radius of r units
!
C =!
" d or C = 2" r
!
!
.
Circles
Notes 10-1 p6
Find the exact circumference of
Find the exact circumference of
.
Notes 10-2 p3
Objectives:
! Recognize major arcs, minor arcs, semicircles, and central angles and
their measures.
! Find arc length.
A CENTRAL ANGLE has the center of the circle as its VERTEX, and
its SIDES contain two RADII of the circle.
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Notes 10-2 p4
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Notes 10-2 p5
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RV is a diameter of
!
Notes 10-2 p6
ARC LENGTH:
ARC LENGTH is DIFFERENT from ARC MEASURE.
An ARC is part of the CIRCLE and is measured in DEGREES.
ARC LENGTH is part of the CIRCUMFERENCE.
Use a PROPORTION to find the ARC LENGTH.
degree measure of arc A
arc length
l
=
degree measure of circle 360 circumference C
!
A
l
=
360 2"r
!
A
•C = l
360
Name___________________________________________________________Notes 10-3 p.1
Date_______________
Objectives:
! Recognize and use relationships between arcs and chords.
! Recognize and use relationships between chords and diameters.
In a circle or in ____________ circles, _____ minor arcs are
__________ if and only if their ____________ __________ are
congruent.
.
.
In a circle, if a _____________ (or __________) is _______________
to a chord, then it ___________ the ___________ and its ________.
In a circle or in ____________ circles, _____ ___________ are
_______________ if and only if they are _______________ from
the ____________.
A polygon is ____________ in a circle if each of its ___________
lie on the circle. ABCD is an ___________ polygon (inside).
A circle is ________________about a polygon if the
circle contains all the _____________of the polygon.
Circle E is ________________ about the polygon (outside)
.
.
Notes 10-3 p.2
Objectives:
! Recognize and use relationships between arcs and
chords.
! Recognize and use relationships between chords and
diameters.
In a circle or in congruent circles, two
minor arcs are congruent if and only
if their corresponding chords are
congruent.
In a circle, if a diameter (or radius) is
perpendicular to a chord, then it
bisects the chord and its arc.
In a circle or in congruent circles, two
chords are congruent if and only if
they are equidistant from the center.
.
.
.
Notes 10-3 p.3
A
A polygon is inscribed in a circle if
each of its vertices lie on the circle.
ABCD is an inscribed polygon
(inside).
E
D
.
C
A circle is circumscribed about a polygon if the circle
contains all the vertices of the polygon. Circle E is
circumscribed about the polygon (outside)
Circle W has a radius of 10
centimeters. Radius WL is
perpendicular to chord HK , which
is 16 centimeters long.
!
! mMK .
If mHL " 53, find
Find JL.
!
.
B
Notes10-4 p1
Objectives:
! Find measures of inscribed angles.
! Find measures of angles of inscribed polygons.
An angle that has its VERTEX on the circle and its SIDES
contained in CHORDS of the circle is an INSCRIBED angle.
An INTERCEPTED arc is an arc whose
endpoints lie on an inscribed angle.
32˚
INSCRIBED Angle Theorem: If an angle is INSCRIBED in a
circle, then the measure of the angle EQUALS ONE-HALF the
measure of its INTERCEPTED ARC (or the measure of the
INTERCEPTED arc is TWICE the measure of the
INSCRIBED angle).
Notes10-4 p2
If TWO inscribed angles of a circle (or congruent circles)
INTERCEPT congruent arcs or the SAME arc, then the angles
are CONGRUENT.
32˚
.
.
If an INSCRIBED angle INTERCEPTS a SEMICIRCLE, the
angle is a RIGHT angle.
58˚
.
Notes10-4 p3
If a QUADRILATERAL is inscribed in a circle, then its
OPPOSITE angles are SUPPLEMENTARY.
.
79˚
86˚
Notes10-4 p4
A
In !P, AC is a diameter,
m"ABE = 40°,
mAB = 86°
D
.
E
B
.P
Find each measure:
1. m " ACB
2. mAD
C
3. mBC
4. m " CAB
5. m " CDB
6. mDC
7. m " DAC
7. 9. m " AED
8. m " DBC
8. 10. m " DEC
!
Notes10-4 p5
In !P, AC is a diameter,
m"ABE = 40°,
mAB = 86°
A
.
D
Find each measure:
.
1. m " ACB
2. mAD
!
E
B
P
3. mBC
4. m " CAB
C
5. m " CDB
!
6. mDC
!
7. m " DAC
7. 9.
8. m " DBC
8. 10. m " DEC
!
!
!
!
m " AED
Notes10-4 p6
Notes10-4 p7
Notes10-4 p8
Notes10-4 p9
Notes10-4 p10
Notes10-4 p11
Objectives:
! Find measures of inscribed angles.
! Find measures of angles of inscribed polygons.
________________ Angles – An angle that has its _______________ on the circle and its
________________ contained in ________________ of the circle.
____________________ Arc –
_________________Angle Theorem: If an angle is ____________________ in
a circle, then the measure of the angle ____________________
_________________ the measure of its __________________ ___________ (or
the measure of the _________________ arc is _________________ the measure
of the ________________ angle).
If _____________ inscribed angles of a circle (or congruent circles)
___________________ congruent arcs or the ____________ arc, then the angles
are _______________.
If an ________________ angle _______________ a _______________, the angle is
a ____________ angle.
If a ______________________ is inscribed in a circle, then its _____________
angles are _________________________ .
Notes10-4 p12
Name___________________________________________10-5 Notes p.1
Date_____________
Objectives:
• Use properties of tangents
• Solve problems involving circumscribed polygons
A _____________ is a __________ or ________
that intersects a ____________ at exactly
_____ __________. This point is called
the ____________ ____ _______________.
If a line is _______________ to a circle then it is __________________ to the
____________ drawn to the ___________ ___ _____________ and if a line is
_______________ to a __________ at its ______________, then it is
_______________ to the circle.
If ____ ______________ from the same ______________ _____________ are
________________ to a circle, then they are ___________________.
Is
EW tangent to !P ?
!
Assume segments that appear tangent are tangent.
Find
2x + 6
3x + 3
!
!
ED
CD
Find the
perimeter
Notes 10-5 p.2
Objectives:
• Use properties of tangents
• Solve problems involving circumscribed polygons
A tangent is a line or ray that
intersects a circle at exactly
one point. This point is called
the point of tangency.
If a line is tangent to a circle then it is perpendicular to
the radius drawn to the point of tangency and if a line
is perpendicular to a radius at its endpoint, then it is
tangent to the circle.
If 2 segments from the same exterior point are tangent
to a circle, then they are congruent.
Is EW tangent to !D ?
!
Assume segments that appear tangent are tangent.
Find
2x + 6
3x + 3
!
!
ED
CD
Find the
perimeter
10-6 p.3
A line that intersects a circle in exactly two
points is called a secant.
If a secant and a tangent intersect at the point
of tangency, then the measure of each angle
formed is one-half the measure of its
intercepted arc.
A
.C
B
If two secants intersect in the interior of a circle,
then the measure of an angle formed is onehalf the sum of the measures of the arcs
intercepted by the angle and its vertical angle.
D
G
E
H
F
10-6 p.4
If two secants, a secant and a tangent, or two
tangents intersect in the exterior of a circle,
then the measure of the angle formed is onehalf the positive difference of the measures of
the intercepted arcs.
two secants
a secant and a tangent
X
T
J
S
K
Z
two tangents
L
Y
M
O
P
10-6 p.5
VERTEX ANGLE
EQUALS
arc
ON
=
2
arc + arc
IN
=
2
OUT
arc " arc
!
=
2
100˚
!
!
D
95˚
F
v˚
w˚
C
E
z˚
y˚
50˚
B x˚
A
Notes 10-7 p.3
Objectives:
! Find the measure of segments that intersect in
the interior of a circle.
! Find measures of segments that intersect in the
exterior of a circle.
A
If two chords intersect in a
circle, then the products of
the measures of the
segments of the chords are
equal.
C
P
D
B
(PA)(PB) = (PC)(PD)
If two secant segments are drawn to a circle from an
exterior point, then the product of the measures of one
secant segment
B
and its external secant
segment is equal to the
A
product of the measures
D
of the other secant
C
segment and its external
secant segment.
(PA)(PB) = (PC)(PD)
P
Notes 10-7 p.4
B
If a tangent segment and a
secant segment are drawn to
a circle from an exterior point,
then the square of the measure
of the tangent segment is equal
to the product of the measures
of the secant segment and its
external secant segment.
C
(PA)(PB) = PC2
P
Solve for x and y:
A
6
D
B 3
x
8
E
12
G
F
A
y
C
Notes 10-8 p.2
Objectives:
! Write the equation of a circle.
! Graph a circle on the coordinate plane.
An equation for a circle with center at (h, k) and
radius of r units is
(x " h) 2 + (y " k) 2 = r 2
Write equation of the circle:
!
center at (–2, 4); diameter = 4
center at (–12, –1); r = 8
center at (3, –3) d = 12
Notes 10-8 p.3
Diameter that has endpoints at (–7, –2) and
(–15, 6).
Notes 10-8 p.4
Graph
(x " 3) 2 + y 2 = 16
!
Graph
!
(x " 2) 2 + (y + 3) 2 = 4