This document discusses circle geometry and contains the following key points: - It defines important parts of a circle like radii, chords, diameters, and arcs. - It establishes theorems relating radii and diameters, congruent arcs and angles, congruent chords and arcs, and the relationships between chords, arcs, and diameters. - It describes how to calculate arc measures, circumference, area of circles and sectors, and properties of inscribed angles and polygons.

1. CIRCLE GEOMETRY
Jyoti Vaid

2. Circles
 11.1 Parts of a Circle
 11.4 Inscribed Polygons
 11.3 Arcs and Chords
 11.2 Arcs and Central Angles
 11.6 Area of a Circle
 11.5 Circumference of a Circle

3. Parts of a Circle
A circle is a special type of geometric figure.
All points on a circle are the same distance from a ___________.center point
O
B
A
The measure of OA and OB are the same; that is, OA OB

4. Parts of a Circle
There are three kinds of segments related to circles.
A ______ is a segment whose endpoints are the center of the circle and a
point on the circle.
radius
A _____ is a segment whose endpoints are on the circle.chord
A ________ is a chord that contains the centerDiameter

5. Parts of a Circle
Segments
of
Circles
K
A
diameterchordradius
R
K
J
K
T
G
Kof
radiusaisKA
Kof
chordaisJR
Kof
diameteraisGT
From the figures, you can not that the diameter is a special type of CHORD
that passes through the center.

6. Theorem
11-1
Theorem
11-2
Parts of a Circle
All radii of a circle are Congruent .
P
R
S
T
G
PTPSPGPR
The measure of the diameter d of a circle is twice the
measure of the radius r of the circle.
r
or
d
rd
2
1
2

7. S
Parts of a Circle
Because all circles have the same shape, any two circles are similar.
However, two circles are congruent if and only if (iff) their radii are _________.congruent
Two circles are concentric if they meet the following three requirements:
Circle R with radius RT and
circle R with radius RS are
concentric circles.
 They lie in the same plane.
 They have the same center.
 They have radii of different lengths.
R
T

8. T
S
Arcs and Central Angles
A ____________ is formed when the two sides of an angle meet at the center
of a circle.
central angle
R
central
angle
Each side intersects a point on the circle, dividing it into arcs that are curved lines.
There are three types of arcs:
A _________ is part of the circle in the interior of
the central angle with measure less than 180 .
minor arc
A _________ is part of the circle in the exterior of
the central angle.
major arc
__________ are congruent arcs whose endpoints
lie on the diameter of the circle.
Semicircles

9. K
P
G
R
Arcs and Central Angles
Types
of
Arcs
semicircle PRTmajor arc PRGminor arc PG
K
P
G
180PGm 180PRGm 180PRTm
Note that for circle K, two letters are used to name the minor arc, but three letters are
used to name the major arc and semicircle. These letters for naming arcs help us
trace the set of points in the arc. In this way, there is no confusion about which arc
is being considered.
K
T
P
R
G

10. Arcs and Central Angles
Depending on the central angle, each type of arc is measured in the
following way.
Definition
of Arc
Measure
1) The degree measure of a minor arc is the degree measure
of its central angle.
2) The degree measure of a major arc is 360 minus the degree
measure of its central angle.
3) The degree measure of a semicircle is 180.

11. Arcs and Central Angles
In P, find the following measures:
P
H
AM
T
46°
80°
MAm = APM
MAm = 46
ATmAPT
THMm = 360 – ( MPA + APT)
APT = 80
THMm = 360 – (46 + 80 )
THMm = 360 – (126 )
THMm = 234

12. Arcs and Central Angles
P
H
AM
T
46°
80°
In P, AM and AT are examples of Adjacent Arcs .
Adjacent arcs have exactly one point in common.
For AM and AT, the common point is __.A
Adjacent arcs can also be added.
Postulate
11-1
Arc
Addition
Postulate
The sum of the measures of two adjacent arcs is the measure
of the arc formed by the adjacent arcs.
C
QP
R
If Q is a point of PR, then
mPQ + mQR mPQR

13. Arcs and Central Angles
B
Suppose there are two concentric circles
S60°
C
A
D
with ASD forming two minor arcs,
BC and AD.
Are the two arcs congruent?
The arcs are in circles with different radii, so they have different lengths.
However, in a circle, or in congruent circles, two arcs are congruent if they
have the same measure.
Although BC and AD each measure 60 , they are not congruent.

14. Arcs and Central Angles
Theorem
11-3
In a circle or in congruent circles, two minor arcs are congruent
if and only if (iff) their corresponding central angles are
congruent.
60° 60°
Z
Y
X
W
Q
WX YZ
iff
m WQX = m YQZ

15. Arcs and Central Angles
T
W
S
K
M
R
In M, WS and RT are diameters, m WMT = 125, mRK = 14.
Find mRS.
WMT RMS Vertical angles are congruent
WMT = RMS Definition of congruent angles
mWT = mRS Theorem 11-3
125 = mRS Substitution
Find mKS.
KS + RK = RS
KS + 14 = 125
KS = 111
Find mTS.
TS + RS = 180
TS + 125 = 180
TS = 55

16. Arcs and Chords
BD
S
C
P
A
In circle P, each chord joins two points on a circle.
Between the two points, an arc forms along the circle.
vertical angles
By Theorem 11-3, AD and BC are congruent
because their corresponding central angles are
_____________, and therefore congruent.
By the SAS Theorem, it could be shown that
ΔAPD ΔCPB.
Therefore, AD and BC are _________.congruent
The following theorem describes the relationship between two congruent
minor arcs and their corresponding chords.

17. Arcs and Chords
Theorem
11-4
In a circle or in congruent circles, two minor arcs are congruent
if and only if (iff) their corresponding ______ are congruent.
B
D
C
A
chords
AD BC
iff
AD BC

18. Arcs and Chords
A
BC
The vertices of isosceles triangle ABC are located on R.
R
If BA AC, identify all congruent arcs.
BA AC

19. Arcs and Chords
Step 1) Use a compass to draw circle on a
piece of patty paper. Label the
center P. Draw a chord that is not
a diameter. Label it EF.
Step 2) Fold the paper through P so that
E and F coincide. Label this fold
as diameter GH.
E
F
P
G
H
Q1: When the paper is folded, how do the lengths of EG and FG compare?
Q2: When the paper is folded, how do the lengths of EH and FH compare?
Q3: What is the relationship between diameter GH and chord EF?
EG FG
EG FG
They appear to be perpendicular.

20. Arcs and Chords
Theorem
11-5
In a circle, a diameter bisects a chord and its arc if and only if
(iff) it is perpendicular to the chord.
P
R
D
C
B
A
AR BR and AD BD
iff
CD AB
Like an angle, an arc can be bisected.

21. Arcs and Chords
B
C
A
K
D
7
Find the measure of AB in K.
DBAB 2 Theorem 11-5
72AB
14AB
Substitution

22. Arcs and Chords
M
K
L
K
N6
Find the measure of KM in K if ML = 16.
222
MNKNKM Pythagorean Theorem
222
86KM Given; Theorem 11-5
6436
2
KM
100
2
KM
100
2
KM
10KM

23. Inscribed Polygons
A
F
B
C
D E
Some regular polygons can be
constructed by inscribing them in circles.
Inscribe a regular hexagon, labeling
the vertices, A, B, C, D, E, and F.
Construct a perpendicular segment
from the center to each chord.
From our study of “regular polygons,”
we know that the chords
AB, BC, CD, DE, and EF are
_________congruent
From the same study, we also know that all of the perpendicular segments,
called ________, are _________.apothems congruent
Make a conjecture about the relationship between the measure of the chords
and the distance from the chords to the center.
The chords are congruent because the distances from the center of the
circle are congruent.
P

24. Inscribed Polygons
Theorem
11-6
In a circle or in congruent circles, two chords are congruent
if and only if they are __________ from the center.equidistant
L
M
P
C
D
B
A
AD BC
iff
LP PM

25. Circumference of a Circle
In the previous activity, the ratio of the circumference C of a circle to its
diameter d appears to be a fixed number slightly greater than 3, regardless
of the size of the circle.
The ratio of the circumference of a circle to its diameter is always fixed and
equals an irrational number called __ or __.pi π
Thus, ____ = __,
d
C π .or dC
.aswrittenbealsocaniprelationshthe2r,dSince rC 2

26. Circumference of a Circle
Theorem
11-7
Circumference
of a Circle
If a circle has a circumference of C units and a radius of r
units, then C = ____r2 or C = ___d
C
d
r

27. Area of a Circle
The space enclosed inside a circle is its area.
By slicing a circle into equal pie-shaped pieces as shown below, you can
rearrange the pieces into a approximate rectangle.
Note that the length along the top and bottom of this rectangle equals the
_____________ of the circle, ____.circumference r2
So, each “length” of this approximate rectangle is half the circumference,
or __r

28. Area of a Circle
The “width” of the approximate rectangle is the radius r of the circle.
Recall that the area of a rectangle is the product of its length and width.
Therefore, the area of this approximate rectangle is (π r)r or ___.
2
r

29. Area of a Circle
Theorem
11-8
Area
of a Circle
If a circle has an area of A square units and a radius of
r units, then A = ___
2
r
2
rA
r

30. Sectors are a fractional part of a
circle’s area

31. Find the shaded area
8 A = πr² A = 64π
Sector area =
¼ of 64π
64π = 16π
4
90 of circle’s area
360

32. 60°
60 of circle’s area
360
12
Area = πr² A = 144π
Sector area =
A = 1 of 144π
6
144π = 24π
6

33. AREA OF SECTION =
AREA OF SECTOR – AREA OF TRIANGLE
¼ π r² - ½ bh

34. Area of section =
area of sector – area of triangle
¼ π r² - ½ bh
10
A OF = ½∙10∙10=
50
A OF SECTION =
25π - 50
A of circle = 100π
A OF = ¼ 100π =
25π

35. Area of a Circle
Theorem 11-9
Area of a
Sector of a
Circle
If a sector of a circle has an area of A square units, a
central angle measurement of N degrees, and a radius of
r units, then
2
360
r
N
A

36. Inscribed Angle:
An angle whose
vertex is on
the circle and
whose sides
are chords of
the circle

37. Determine whether each angle is an
inscribed angle. Name the intercepted
arc for the angle.
C
L
O
T
1.
YES;
CL

38. Determine whether each angle is an
inscribed angle. Name the intercepted
arc for the angle.
Q
R
K
V
2. NO;
QVR
S

39. 2
ArcdIntercepte
AngleInscribed
1600
800
To find the measure of an inscribed angle…

40. 120
x
What is the value of x?
y
What do we call this type of angle?
How do we solve for y?
The measure of the inscribed angle is HALF the
measure of the inscribed arc!!

41. 72
If two inscribed angles intercept the
same arc, then they are congruent.

42. If all the vertices of a polygon
touch the edge of the circle, the
polygon is INSCRIBED and the
circle is CIRCUMSCRIBED.

43. A circle can be circumscribed around
a quadrilateral if and only if its
opposite angles are supplementary.
A B
CD
180CmAm
180DmBm

44. z
y
110
85
110 + y =180
y = 70
z + 85 = 180
z = 95
Example 8 Find y and z.

45. 180
If a right
triangle is
inscribed in a
circle then the
hypotenuse is
the diameter of
the circle.

46. H
K
G
N
4x – 14 = 90
Example 6
In K, m<GNH = 4x – 14. Find the value of x.
x = 26