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This material is made freely available at www.njctl.org and is intended for the non-commercial use of students and teachers. These materials may not be used for any commercial purpose without the written permission of the owners. NJCTL maintains its website for the convenience of teachers who wish to make their work available to other teachers, participate in a virtual professional learning community, and/or provide access to course materials to parents, students and others.
Click to go to website:www.njctl.org
New Jersey Center for Teaching and Learning
Progressive Mathematics Initiative®
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Geometry
Circles
www.njctl.org
2014-06-03
Slide 3 / 150
Table of Contents
Parts of a CircleAngles & ArcsChords, Inscribed Angles & Polygons
Segments & CirclesEquations of a Circle
Click on a topic to go to that section
Tangents & Secants
Area of a Sector
Slide 5 / 150
A circle is the set of all points in a plane that are a fixed distance from a given point in the plane called the center.
center
Slide 6 / 150
The symbol for a circle is and is named by a capital letter placed by the center of the circle.
.
A
B
(circle A or . A)is a radius of . A
A radius (plural, radii) is a line segment drawn from the center of the circle to any point on the circle. It follows from the definition of a circle that all radii of a circle are congruent.
.is a radius of A).(circle A or
A
.
Slide 7 / 150
A
M
C
R
T
is the diameter of circle A
is a chord of circleAA chord is a segment that has its endpoints on the circle.
A diameter is a chord that goes through the center of the circle. All diameters of a circle are congruent.
What are the radii in this diagram?
Ans
wer
Slide 8 / 150
The relationship between the diameter and the radius
A
The measure of the diameter, d, is twice the measure of the radius, r.
Therefore, orM
C
T
If then what is the length of ,
In . A
what is the length of
Ans
wer
Slide 9 / 150
1 A diameter of a circle is the longest chord of the circle.True
False
Ans
wer
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2 A radius of a circle is a chord of a circle.
True
False
Ans
wer
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3 Two radii of a circle always equal the length of a diameter of a circle.
True
False
Ans
wer
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4 If the radius of a circle measures 3.8 meters, what is the measure of the diameter?
Ans
wer
Slide 13 / 150
5 How many diameters can be drawn in a circle?
A 1
B 2
C 4D infinitely many
Ans
wer
Slide 14 / 150
A secant of a circle is a line that intersects the circle at two points.A
B
D
E k
l
line l is a secant of this circle.
A tangent is a line in the plane of a circle that intersects the circle at exactly one point (the point of tangency).
line k is a tangent D is the point of tangency.
tangent ray, , and the tangent segment, , are also called tangents. They must be part of a tangent line.
Note: This is not a tangent ray.
Slide 15 / 150
COPLANAR CIRCLES are two circles in the same plane which intersect at 2 points, 1 point, or no points.
Coplanar circles that intersects in 1 point are called tangent circles. Coplanar circles that have a common center are called concentric.
2 points tangent circles
1 point
concentric circles
....
.
no points
Slide 16 / 150
A Common Tangent is a line, ray, or segment that is tangent to 2 coplanar circles.
Internally tangent(tangent line
passes between them)
Externally tangent(tangent line does not pass between
them)
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6 How many common tangent lines do the circles have?
Ans
wer
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7 How many common tangent lines do the circles have?
Ans
wer
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8 How many common tangent lines do the circles have?
Ans
wer
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9 How many common tangent lines do the circles have?
Ans
wer
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Using the diagram below, match the notation with the term that best describes it:
A
C
D
E
F
G.
.
..
. .
B.
centerradiuschord
diametersecanttangent point of tangency
common tangent
Ans
wer
Slide 23 / 150
An ARC is an unbroken piece of a circle with endpoints on the circle.
..
A
B
Arc of the circle or AB
Arcs are measured in two ways:1) As the measure of the central angle in degrees2) As the length of the arc itself in linear units
(Recall that the measure of the whole circle is 360o.)
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A central angle is an angle whose vertex is the center of the circle.
M
AT
HS. .In , is the central
angle.
Name another central angle.
Ans
wer
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M
AT
HS. .minor arc MA
If is less than 1800, then the points on that lie in the interior of form the minor arc with endpoints M and H.
Name another minor arc.
MAHighlight
Ans
wer
Slide 26 / 150
M
AT
HS. .major arc
Points M and A and all points of exterior to form a major arc, Major arcs are the "long way" around the circle.
Major arcs are greater than 180o. Highlight
Major arcs are named by their endpoints and a point on the arc.
Name another major arc.
MSA
MSA
Ans
wer
Slide 27 / 150
M
AT
HS. . minor arc
A semicircle is an arc whose endpoints are the endpoints of the diameter.
MAT is a semicircle. Highlight the semicircle.
Semicircles are named by their endpoints and a point on the arc.
Name another semicircle.
Ans
wer
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The measure of a minor arc is the measure of its central angle.The measure of the major arc is 3600 minus the measure of the central angle.
Measurement By A Central Angle
A
B
D.400
G
400
3600 - 400 = 3200
Slide 29 / 150
The Length of the Arc Itself (AKA - Arc Length)
Arc length is a portion of the circumference of a circle.
Arc Length Corollary - In a circle, the ratio of the length of a given arc to the circumference is equal to the ratio of the measure of the arc to 3600.
C
A
T
r
arc length of =3600
CT CT
CT CTarc length of =3600
.or
Slide 30 / 150
C
A
T
8 cm
600
EXAMPLE
In , the central angle is 600 and the radius is 8 cm.Find the length of
ACT
Ans
wer
Slide 31 / 150
EXAMPLE
S
A
Y
4.19 in
400
AIn , the central angle is 400 and the length of is 4.19 in. Find the circumference of A.
SYA.
In , the central angle is 400 and the length of is 4.19 in. Find the circumference of
SYA
Ans
wer
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10 In circle C where is a diameter, find
1350
A
C
B
D
15 in
Ans
wer
Slide 33 / 150
11 In circle C, where is a diameter, find
1350
A
C
B
D
15 in
Ans
wer
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12 In circle C, where is a diameter, find
1350
A
C
B
D
15 in
Ans
wer
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13 In circle C can it be assumed that AB is a diameter?
Yes
No 1350
A
C
B
D Ans
wer
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14 Find the length of
450
A
C3 cm
B
Ans
wer
Slide 37 / 150
15 Find the circumference of circle T.
T750
6.82 cm
Ans
wer
Slide 38 / 150
1400
16 In circle T, WY & XZ are diameters. WY = XZ = 6.
If XY = , what is the length of YZ?
A
B
C
D
T
W
Y
X
Z
Ans
wer
Slide 39 / 150
Adjacent arcs: two arcs of the same circle are adjacent if they have a common endpoint.
Just as with adjacent angles, measures of adjacent arcs can be added to find the measure of the arc formed by the adjacent arcs.
ADJACENT ARCS
.
..
C
A
T
+=
Slide 40 / 150
EXAMPLEA result of a survey about the ages of people in a city are shown. Find the indicated measures.
>65
45-64
15-17
17-44
S
U
V
R
300
900
800
600
1000
T
1.
2.
3.
4.
Ans
wer
Slide 41 / 150
Match the type of arc and it's measure to the given arcs below:
1200
800 600
T
SR
Q
minor arc major arc semicircle
1200 240018001600800
Teac
her N
otes
Slide 42 / 150
CONGRUENT CIRCLES & ARCS· Two circles are congruent if they have the same radius.· Two arcs are congruent if they have the same measure and they are arcs of the same circle or congruent circles.
C
D E
F550 550
R
S
T
U
& because they are in the same circle and
have the same measure, but are not congruent because they are arcs of circles that are not congruent.
Slide 43 / 150
17
True
False1800
700
400
A
B
C
D
Ans
wer
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18
True
False 850
M
N
L
P
Ans
wer
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90019 Circle P has a radius of 3 and has a measure
of . What is the length of ?
A
B
C
D
P
A
B
Ans
wer
Slide 46 / 150
20 Two concentric circles always have congruent radii.
True
False
Ans
wer
Slide 47 / 150
21 If two circles have the same center, they are congruent.
True
False
Ans
wer
Slide 48 / 150
22 Tanny cuts a pie into 6 congruent pieces. What is the measure of the central angle of each piece?
Ans
wer
Slide 50 / 150
Lab - Properties of Chords
Click on the link below and complete the labs before the Chords lesson.
Slide 51 / 150
is the arc of
When a minor arc and a chord have the same endpoints, we call the arc The Arc of the Chord.
.C
P
Q
**Recall the definition of a chord - a segment with endpoints on the circle.
Slide 52 / 150
THEOREM:
In a circle, if one chord is a perpendicular bisector of another chord, then the first chord is a diameter.
T
P
SQ E
is the perpendicular bisector of .
Therefore, is a diameter of the circle.
Likewise, the perpendicular bisector of a chord of a circle passes through the center of a circle.
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THEOREM:If a diameter of a circle is perpendicular to a chord, then the diameter bisects the chord and its arc.
AC
E
SX.
is a diameter of the circle and is perpendicular to chord
Therefore,
Slide 54 / 150
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
D
if f
*iff stands for "if and only if"
Slide 55 / 150
If , then point Y and any line segment, or ray, that contains Y, bisects
BISECTING ARCS
C
X
Z
Y
Slide 56 / 150
Find:,, and
EXAMPLE
A
BC
D
E
. (9x)0
(80 - x)0
and, ,Find:
Ans
wer
Slide 57 / 150
THEOREM:In the same circle, or congruent circles, two chords are congruent if and only if they are equidistant from the center.
.C
G
DEA
FB
iff
Slide 58 / 150
EXAMPLE
Given circle C, QR = ST = 16. Find CU.
.Q
R
S
T
U
V
2x
5x - 9C
Since the chords QR & ST are congruent, they are equidistant from C. Therefore,
Ans
wer
Slide 59 / 150
23 In circle R, and . FindA
B
C
D
R.1080
Ans
wer
Slide 60 / 150
24 Given circle C below, the length of is:
A 5
B 10
C 15
D 20
D B
F
C.10
A
Ans
wer
Slide 61 / 150
25 Given: circle P, PV = PW, QR = 2x + 6, and ST = 3x - 1. Find the length of QR.
A 1
B 7C 20
D 8
R
SQ
T
P
W
.V
Ans
wer
Slide 62 / 150
26 AH is a diameter of the circle.
True
False
A
S
H
M
3
3
5
T
Ans
wer
Slide 63 / 150
INSCRIBED ANGLESD
OG
Inscribed angles are angles whose vertices are in on the circle and whose sides are chords of the circle.
The arc that lies in the interior of an inscribed angle, and has endpoints on the angle, is called the intercepted arc.
is an inscribed angle and is its intercepted arc.
Lab - Inscribed Angles
Click on the link below and complete the lab.
Slide 64 / 150
THEOREM:The measure of an inscribed angle is half the measure of its intercepted arc.
C
A
T
Slide 65 / 150
EXAMPLE
Q R
T S
P.500
480
Find and
Ans
wer
Slide 66 / 150
THEOREM:If two inscribed angles of a circle intercept the same arc, then the angles are congruent.
D
C
B
Asince they both intercept
Slide 67 / 150
In a circle, parallel chords intercept congruent arcs.
O
B
.A
DC
In circle O, if , then, thenIn circle O, if
Slide 68 / 150
27 Given circle C below, find
D E
C
A B
. 1000
350 Ans
wer
Slide 69 / 150
28 Given circle C below, find
D E
C
A B
. 1000
350
Ans
wer
Slide 70 / 150
29 Given the figure below, which pairs of angles are congruent?
A
B
C
D
RS
U
T
Ans
wer
Slide 71 / 150
30 Find
X
Y
Z
P.
Ans
wer
Slide 72 / 150
31 In a circle, two parallel chords on opposite sides of the center have arcs which measure 1000 and 1200. Find the measure of one of the arcs included between the chords.
Ans
wer
Slide 73 / 150
32 Given circle O, find the value of x.
.O
A B
C D
x
300
Ans
wer
Slide 74 / 150
33 Given circle O, find the value of x.
.O
A B
C D
x
1000
350
Ans
wer
Slide 75 / 150
In the circle below, and Find
, and
Try This
P
S
1
2
3
4
Q
T
Ans
wer
Slide 76 / 150
INSCRIBED POLYGONS
A polygon is inscribed if all its vertices lie on a circle.
.
.
.
inscribed triangle
.
.
.
.
inscribed quadrilateral
Slide 77 / 150
THEOREM:If a right triangle is inscribed in a circle, then the hypotenuse is a diameter of the circle.
A
L
G
x.iff AC is a diameter of the circle.
Slide 78 / 150
THEOREM:A quadrilateral can be inscribed in a circle if and only if its opposite angles are supplementary.
N
E
R
AC. N, E, A, and R lie on circle C iff
Slide 79 / 150
EXAMPLE
Find the value of each variable:
2a
2a4b
2bL
K
J
M
Ans
wer
Slide 80 / 150
34 The value of x is
A
B
C
D
1500
980
1120
1800
C
B
A
Dx
y
680
820
Ans
wer
Slide 81 / 150
35 In the diagram, is a central angle and . What is ?
150
300
600
1200
A
B
C
D
.B
A
DC
1200
600
300
150
Ans
wer
Slide 82 / 150
36 What is the value of x?
A 5
B 10
C 13
D 15
E
F G
(12x + 40)0
(8x + 10)0
Ans
wer
Slide 84 / 150
**Recall the definition of a tangent line: A line that intersects the circle in exactly one point.
THEOREM:In a plane, a line is tangent to a circle if and only if the line is perpendicular to a radius of the circle at its endpoint on the circle (the point of tangency).
..X
B
l
lLine is tangent to circle X iff would be the point of tangency.BBLine is tangent to circle X iff would be the point of tangency.
l l
Lab - Tangent Lines
Click on the link below and complete the lab.
Slide 85 / 150
Verify A Line is Tangent to a Circle
.T
P
S
35
37
12}
Given: is a radius of circle PIs tangent to circle P?
Ans
wer
Slide 86 / 150
Finding the Radius of a Circle
.A
C
B
r
r
50 ft
80 ft
If B is a point of tangency, find the radius of circle C.
Ans
wer
Slide 87 / 150
THEOREM:Tangent segments from a common external point are congruent.
R
A
T
P.
If AR and AT are tangent segments, then
Slide 88 / 150
EXAMPLE
Given: RS is tangent to circle C at S and RT is tangent to circle C at T. Find x.
S
R
T
C.28
3x + 4 Ans
wer
Slide 89 / 150
37 AB is a radius of circle A. Is BC tangent to circle A?
Yes
No
.
B C
A
60
25
67
}
Ans
wer
Slide 90 / 150
38 S is a point of tangency. Find the radius r of circle T.
A 31.7
B 60
C 14
D 3.5
.T
SR
r
r
48 cm
36 cm
Ans
wer
Slide 91 / 150
39 In circle C, DA is tangent at A and DB is tangent at B. Find x.
A
D
B
C.25
3x - 8
Ans
wer
Slide 92 / 150
40 AB, BC, and CA are tangents to circle O. AD = 5, AC= 8, and BE = 4. Find the perimeter of triangle ABC.
.
B
EF
AC D
O
Ans
wer
Slide 93 / 150
Tangents and secants can form other angle relationships in circle. Recall the measure of an inscribed angle is 1/2 its intercepted arc. This can be extended to any angle that has its vertex on the circle. This includes angles formed by two secants, a secant and a tangent, a tangent and a chord, and two tangents.
Slide 94 / 150
A Tangent and a 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.
..
.
A
M
R2 1
Slide 95 / 150
A Tangent and a Secant, Two Tangents, and Two Secants
THEOREM:If a tangent and a secant, two tangents, or two secants intersect outside a circle, then the measure of the angle formed is half the difference of its intercepted arcs.
A
B
C
1
a tangent and a secant
PQ
M
2
.
two tangents two secants
W
X
YZ
3
Slide 96 / 150
THEOREM:If two chords intersect inside a circle, then the measure of each angle is half the sum of the intercepted arcs by the angle and vertical angle.
M A
H T
12
Slide 97 / 150
EXAMPLE
Find the value of x.
D
C
AB
x0 7601780
Ans
wer
Slide 98 / 150
EXAMPLE
Find the value of x.1300
x0
1560
Ans
wer
Slide 99 / 150
41 Find the value of x.
C
H
DFx0
780
420
E
Ans
wer
Slide 100 / 150
42 Find the value of x.
340
(x + 6)0
(3x - 2)0
Ans
wer
Slide 101 / 150
43 Find
A
B
650
Ans
wer
Slide 102 / 150
44 Find
12600
Ans
wer
Slide 103 / 150
45 Find the value of x.
x122.50
450
Ans
wer
Slide 104 / 150
2470
A
B
x0
To find the angle, you need the measure of both intercepted arcs. First, find the measure of the minor arc . Then we can calculate the measure of the angle . x0
Ans
wer
Slide 105 / 150
46 Find the value of x.
Students type their answers here
2200
x0
Ans
wer
Slide 106 / 150
47 Find the value of x. Students type their answers here
x01000 A
nsw
er
Slide 107 / 150
48 Find the value of x Students type their answers here
x0
500
Ans
wer
Slide 108 / 150
49 Find the value of x. Students type their answers here
1200
(5x + 10)0
Ans
wer
Slide 109 / 150
50 Find the value of x.
(2x - 30)0
300 x
Ans
wer
Slide 111 / 150
THEOREM:If two chords intersect inside a circle, then the products of the measures of the segments of the chords are equal.
A C
D B
E
Slide 112 / 150
EXAMPLE
Find the value of x.
5
5
x
4
Ans
wer
Slide 113 / 150
EXAMPLEFind ML & JK.
x + 2x +
4
x x + 1
M K
J
L
Ans
wer
Slide 114 / 150
51 Find the value of x.
189
16
x
Ans
wer
Slide 115 / 150
52 Find the value of x.
A -2
B 4
C 5
D 6 x2
2x + 6
x
Ans
wer
Slide 116 / 150
THEOREM:If two secant segments are drawn to a circle from an external point, then the product of the measures of one secant segment and its external secant segment equals the product of the measures of the other secant segment and its external secant segment.
A
B
E C D
Slide 117 / 150
EXAMPLE
Find the value of x.
9 6
x 5
Ans
wer
Slide 118 / 150
53 Find the value of x.
3
x + 2x + 1
x - 1
Ans
wer
Slide 119 / 150
54 Find the value of x.
x + 4
x - 2
5
4
Ans
wer
Slide 120 / 150
THEOREM:If a tangent segment and a secant segment are drawn to a circle from an external 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.
A
E CD
Slide 121 / 150
EXAMPLEFind RS.
R S
Q
T
16
x 8
Ans
wer
Slide 122 / 150
55 Find the value of x.
1
x
3
Ans
wer
Slide 123 / 150
56 Find the value of x.
x12
24
Ans
wer
Slide 125 / 150
y
x
r(x, y)
Let (x, y) be any point on a circle with center at the origin and radius, r. By the Pythagorean Theorem,
x2 + y2 = r2
This is the equation of a circle with center at the origin.
Slide 126 / 150
EXAMPLE
Write the equation of the circle.
4 Ans
wer
Slide 127 / 150
For circles whose center is not at the origin, we use the distance formula to derive the equation.
.(x, y)
(h, k)
r
This is the standard equation of a circle.
Slide 128 / 150
EXAMPLE
Write the standard equation of a circle with center (-2, 3) & radius 3.8.
Ans
wer
Slide 129 / 150
EXAMPLE
The point (-5, 6) is on a circle with center (-1, 3). Write the standard equation of the circle.
Ans
wer
Slide 130 / 150
EXAMPLE
The equation of a circle is (x - 4)2 + (y + 2)2 = 36. Graph the circle.
We know the center of the circle is (4, -2) and the radius is 6.
..
..
First plot the center and move 6 places in each direction.
Then draw the circle.
Slide 131 / 150
57 What is the standard equation of the circle below?
A
B
C
D
x2 + y2 = 400
(x - 10)2 + (y - 10)2 = 400
(x + 10)2 + (y - 10)2 = 400
(x - 10)2 + (y + 10)2 = 40010
Ans
wer
Slide 132 / 150
58 What is the standard equation of the circle?A
B
C
D
(x - 4)2 + (y - 3)2 = 9
(x + 4)2 + (y + 3)2 = 9
(x + 4)2 + (y + 3)2 = 81
(x - 4)2 + (y - 3)2 = 81
Ans
wer
Slide 133 / 150
59 What is the center of (x - 4)2 + (y - 2)2 = 64?
A (0,0)
B (4,2)
C (-4, -2)
D (4, -2) Ans
wer
Slide 134 / 150
60 What is the radius of (x - 4)2 + (y - 2)2 = 64?
Ans
wer
Slide 135 / 150
61 The standard equation of a circle is (x - 2)2 + (y + 1)2 = 16. What is the diameter of the circle?
A 2
B 4
C 8
D 16
Ans
wer
Slide 136 / 150
62 Which point does not lie on the circle described by the equation (x + 2)2 + (y - 4)2 = 25?
A (-2, -1)
B (1, 8)
C (3, 4)
D (0, 5)
Ans
wer
Slide 138 / 150
A sector of a circle is the portion of the circle enclosed by two radii and the arc that connects them.
A
B
C
Minor Sector
Major Sector
Slide 139 / 150
63 Which arc borders the minor sector?
A
B A
BC
D
Ans
wer
Slide 140 / 150
64 Which arc borders the major sector?
A
B
W
X
YZ
Ans
wer
Slide 141 / 150
Lets think about the formula...The area of a circle is found by
We want to find the area of part of the circle, so the formula for the area of a sector is the fraction of the circle multiplied by the area of the circle
When the central angle is in degrees, the fraction of the circle is out of the total 360 degrees.
Slide 142 / 150
Finding the Area of a Sector1. Use the formula: when θ is in degrees
450
AB
C
r= 3 Ans
wer
Slide 143 / 150
Example:Find the Area of the major sector.
C
A
T
8 cm
600
Ans
wer
Slide 144 / 150
65 Find the area of the minor sector of the circle. Round your answer to the nearest hundredth.
C
A
T5.5 cm 300
Ans
wer
Slide 145 / 150
66 Find the Area of the major sector for the circle. Round your answer to the nearest thousandth.
C
A
T
12 cm
850
Ans
wer
Slide 146 / 150
67 What is the central angle for the major sector of the circle?
C
A
G
15 cm
1200
Ans
wer
Slide 147 / 150
68 Find the area of the major sector. Round to the nearest hundredth.
C
A
G
15 cm
1200
Ans
wer
Slide 148 / 150
69 The sum of the major and minor sectors' areas is equal to the total area of the circle.
True
False
Ans
wer
Slide 149 / 150
70 A group of 10 students orders pizza. They order 5 12" pizzas, that contain 8 slices each. If they split the pizzas equally, how many square inches of pizza does each student get?
Ans
wer
Slide 150 / 150
71 You have a circular sprinkler in your yard. The sprinkler has a radius of 25 ft. How many square feet does the sprinkler water if it only rotates 120 degrees?
Ans
wer
