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New Jersey Center for Teaching and Learning Progressive Mathematics Initiative. - PowerPoint PPT Presentation
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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
Geometry
Circles
www.njctl.org
2014-03-31
Table of Contents
Parts of a Circle
Angles & Arcs
Chords, Inscribed Angles & Polygons
Segments & Circles
Equations of a Circle
Click on a topic to go to that section
Tangents & Secants
Area of a Sector
Parts of a Circle
Return to the table of contents
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
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
.
A
M
C
R
T
is the diameter of circle A
is a chord of circle AA 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?
Answ
er
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
Answ
er
1 A diameter of a circle is the longest chord of the circle.
True
False
Answ
er
2 A radius of a circle is a chord of a circle.
True
False
Answ
er
3 Two radii of a circle always equal the length of a diameter of a circle.
True
False
Answ
er
4 If the radius of a circle measures 3.8 meters, what is the measure of the diameter?
Answ
er
5 How many diameters can be drawn in a circle?
A 1
B 2
C 4
D infinitely many
Answ
er
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.
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 circles1 point
concentric circles
..
...
no points
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)
6 How many common tangent lines do the circles have?
Answ
er
7 How many common tangent lines do the circles have?
Answ
er
8 How many common tangent lines do the circles have?
Answ
er
9 How many common tangent lines do the circles have?
Answ
er
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
Answ
er
Angles & Arcs
Return to the table of contents
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.)
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.
Answ
er
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
Answ
er
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
Answ
er
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.
Answ
er
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
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
C
A
T
8 cm
600
EXAMPLE
In , the central angle is 600 and the radius is 8 cm.Find the length of
A
CT
Answ
er
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
Answ
er
10 In circle C where is a diameter, find
1350
A
C
B
D
15 in
Answ
er
11 In circle C, where is a diameter, find
1350
A
C
B
D
15 in
Answ
er
12 In circle C, where is a diameter, find
1350
A
C
B
D
15 in
Answ
er
13 In circle C can it be assumed that AB is a diameter?
Yes
No 1350
A
C
B
D
Answ
er
14 Find the length of
450
A
C3 cm
B
Answ
er
15 Find the circumference of circle T.
T
750
6.82 cm
Answ
er
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
Answ
er
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
+=
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
300900
800600
1000
T
1.
2.
3.
4.
Answ
er
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
Not
es
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.
17
True
False1800
700400
A
B
C
D
Answ
er
18
True
False 850
M
N
L
P
Answ
er
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
Answ
er
20 Two concentric circles always have congruent radii.
True
False
Answ
er
21 If two circles have the same center, they are congruent.
True
False
Answ
er
22 Tanny cuts a pie into 6 congruent pieces. What is the measure of the central angle of each piece?
Answ
er
Chords, Inscribed Angles & Polygons
Return to the table of contents
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.
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.
THEOREM:If a diameter of a circle is perpendicular to a chord, then the diameter bisects the chord and its arc.
A
C
E
S
X
. is a diameter of the circle and is perpendicular to chord
Therefore,
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
iff
*iff stands for "if and only if"
If , then point Y and any line segment, or ray, that contains Y, bisects
BISECTING ARCS
C
X
Z
Y
Find:
,, and
EXAMPLE
A
BC
D
E
. (9x)0
(80 - x)0
and, ,
Find:
Answ
er
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
EXAMPLE
Given circle C, QR = ST = 16. Find CU.
.Q
R
S
T
U
V
2x
5x - 9
C
Since the chords QR & ST are congruent, they are equidistant from C. Therefore,
Answ
er
23 In circle R, and . FindA
B
C
D
R.1080
Answ
er 1080
24 Given circle C below, the length of is:
A 5
B 10
C 15
D 20
D B
F
C.10
A
Answ
er D
25 Given: circle P, PV = PW, QR = 2x + 6, and ST = 3x - 1. Find the length of QR.
A 1
B 7
C 20
D 8
R
SQ
T
P
W
.V
Answ
er C
26 AH is a diameter of the circle.
True
False
A
S
H
M
3
3
5
T
Answ
er
False
INSCRIBED ANGLES
D
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.
THEOREM:The measure of an inscribed angle is half the measure of its intercepted arc.
C
A
T
EXAMPLE
Q R
T S
P. 500
480
Find and
Answ
er
THEOREM:If two inscribed angles of a circle intercept the same arc, then the angles are congruent.
D
C
B
Asince they both intercept
In a circle, parallel chords intercept congruent arcs.
O
B
.A
DC
In circle O, if , then, thenIn circle O, if
27 Given circle C below, find
D E
C
A B
. 1000
350 Answ
er
28 Given circle C below, find
D E
C
A B
. 1000
350
Answ
er 1100
29 Given the figure below, which pairs of angles are congruent?
A
B
C
D
RS
U
T
Answ
er
30 Find
X
Y
Z
P.
Answ
er 900
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.
Answ
er
32 Given circle O, find the value of x.
.O
A B
C D
x
300
Answ
er 1200
33 Given circle O, find the value of x.
.O
A B
C D
x
1000
350
Answ
er 1200
In the circle below, and Find
, and
Try This
P
S
1
2
3
4
Q
T
Answ
er
INSCRIBED POLYGONS
A polygon is inscribed if all its vertices lie on a circle.
.
.
.
inscribed triangle
.
.
.
.
inscribed quadrilateral
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.
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
EXAMPLE
Find the value of each variable:
2a
2a4b
2bL
K
J
M
Answ
er
34 The value of x is
A
B
C
D
1500
980
1120
1800
C
B
A
Dx
y
680
820
Answ
er
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
Answ
er
36 What is the value of x?
A 5
B 10
C 13
D 15
E
F G
(12x + 40)0
(8x + 10)0
Answ
er
Tangents & Secants
Return to the table of contents
**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
Verify A Line is Tangent to a Circle
.T
P
S
35
37
12}
Given: is a radius of circle P
Is tangent to circle P?
Answ
er
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.
Answ
er
THEOREM:Tangent segments from a common external point are congruent.
R
A
T
P.
If AR and AT are tangent segments, then
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 Answ
er
37 AB is a radius of circle A. Is BC tangent to circle A?
Yes
No
.B C
A
60
25
67
}
Answ
er
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
Answ
er
39 In circle C, DA is tangent at A and DB is tangent at B. Find x.
A
D
B
C.25
3x - 8
Answ
er
40 AB, BC, and CA are tangents to circle O. AD = 5, AC= 8, and BE = 4. Find the perimeter of triangle ABC.
.
B
E
F
AC D
O Answ
er
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.
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
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
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
EXAMPLE
Find the value of x.
D
C
A
B
x0 760 1780
Answ
er
EXAMPLE
Find the value of x.
1300
x0
1560
Answ
er
41 Find the value of x.
C
H
DFx0
780
420E
Answ
er
42 Find the value of x.
340(x + 6)0
(3x - 2)0
Answ
er
43 Find
A
B
650
Answ
er
44 Find
12600
Answ
er
45 Find the value of x.
x122.50
450
Answ
er
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
Answ
er
46 Find the value of x.
2200
x0
Answ
er
47 Find the value of x.
x01000 An
swer
48 Find the value of x
x0500
Answ
er
49 Find the value of x.
1200
(5x + 10)0
Answ
er
50 Find the value of x.
(2x - 30)0
300 x
Answ
er
Segments & Circles
Return to the table of contents
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
EXAMPLE
Find the value of x.
5
5
x
4
Answ
er
EXAMPLE
Find ML & JK.
x + 2
x + 4
x x + 1
M K
J
L
Answ
er
51 Find the value of x.
189
16
x
Answ
er
52 Find the value of x.
A -2
B 4
C 5
D 6x
2
2x + 6
x
Answ
er
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
EXAMPLE
Find the value of x.
9 6
x 5
Answ
er
53 Find the value of x.
3
x + 2x + 1
x - 1
Answ
er
54 Find the value of x.
x + 4
x - 2
5
4
Answ
er
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
EXAMPLE
Find RS.
R S
Q
T
16
x 8
Answ
er
55 Find the value of x.
1
x
3
Answ
er
56 Find the value of x.
x12
24
Answ
er
Equations of a Circle
Return to the table of contents
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.
EXAMPLE
Write the equation of the circle.
4 Answ
er
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.
EXAMPLE
Write the standard equation of a circle with center (-2, 3) & radius 3.8.
Answ
er
EXAMPLE
The point (-5, 6) is on a circle with center (-1, 3). Write the standard equation of the circle.
Answ
er
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.
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
Answ
er
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
Answ
er
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) Answ
er
60 What is the radius of (x - 4)2 + (y - 2)2 = 64?
Answ
er
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
Answ
er
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)
Answ
er
Return to the table of contents
Area of a Sector
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
63 Which arc borders the minor sector?
A
B A
BC
D
Answ
er
64 Which arc borders the major sector?
A
B
W
X
YZ
Answ
er
Lets think about the formula...The area of a circle is found byWe 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.
Finding the Area of a Sector1. Use the formula: when θ is in degrees
450
AB
C
r=3
Answ
er
Example:
Find the Area of the major sector.
C
A
T
8 cm
600 Answ
er
65 Find the area of the minor sector of the circle. Round your answer to the nearest hundredth.
C
A
T5.5 cm 300
Answ
er
66 Find the Area of the major sector for the circle. Round your answer to the nearest thousandth.
C
A
T
12 cm
850
Answ
er
67 What is the central angle for the major sector of the circle?
C
A
G
15 cm
1200
Answ
er
68 Find the area of the major sector. Round to the nearest hundredth.
C
A
G
15 cm
1200
Answ
er
69 The sum of the major and minor sectors' areas is equal to the total area of the circle.
True
False
Answ
er
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?
Answ
er
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?
Answ
er