Unit 25
CIRCLES
DEFINITIONS





circle - closed curve in which every point
on the curve is equally distant from a
fixed point called the center
circumference - the length of the curved
line that forms the circle
chord - a straight line segment that joins
two joints on the circle
diameter - a chord that passes through
the center of a circle
radius - a straight line segment that
connects the center of a circle with a
point on the circle
2
DEFINITIONS (Cont)



arc - that part of a circle
between any two points
on the circle. FB is an
arc
tangent - a straight line
that touches the circle
at only one point. DE is
a tangent line
A secant - a straight line
passing through a circle
and intersecting the
circle at two points. AC
is a secant line
B
F
A
•
•
•
•
•
D
•
C
E
3
DEFINITIONS (Cont)




segment - a figure formed by an arc
and the chord joining the end points
of the arc
sector - a figure formed by two radii
and the arc intercepted by the radii
central angle - an angle whose
vertex is at the center of the circle
and whose sides are radii
inscribed angle - an angle in a circle
whose vertex is on the circle and
whose sides are chords
4
CIRCUMFERENCE


The circumference of a circle is
equal to  times the diameter or 2
times the radius
C = d or C = 2r
Find the circumference of a circle
whose diameter is 3 ft. Round your
answer to three significant digits
C = π(3) = 9.42 ft Ans
5
ARC LENGTH


The length of an arc equals the ratio
of the number of degrees of the arc
to 360° times the circumference
Determine the length of a 30° arc
on a circle with a radius of 5 m:
30
2 5m  2.618 m Ans
Arc length 

360
6
CIRCLE POSTULATES




In the same circle or in equal circles, equal
chords subtend (cut off) equal arcs (arcs of
equal length)
In the same circle or in congruent circles,
equal central angles subtend (cut off) equal
arcs
In the same circle or congruent circles, two
central angles have the same ratio as the
arcs that are subtended (cut off) by the
angles
A diameter perpendicular to a chord bisects
the chord and the arcs subtended by the
chord; the perpendicular bisector of a chord
passes through the center of the circle
7
POSTULATE EXAMPLE

Determine the length of arc AB is the
figure below given that CD is 24 cm,
COD = 92, and AOB = 58
C
D
O
A
B
– The postulate “In the same circle, two central
angles have the same ratio as the arcs that are
subtended by the angles” applies here
92 24 cm
thus,


58
AB
– Solving the proportion, arc AB = 15.13 cm Ans
8
CIRCLE TANGENTS AND CHORD
SEGMENTS



A line perpendicular to a radius at its
extremity is tangent to the circle; a tangent
to a circle is perpendicular to the radius at
the tangent point
Two tangents drawn to a circle from a point
outside the circle are equal and make equal
angles with the line joining the point to the
center
If two chords intersect inside a circle, the
product of the two segments of one chord is
equal to the product of the two segments of
the other chord
9
TANGENT AND CHORD
EXAMPLES

Find the value of APO in the figure
below, given that APB = 84:
•A
+O
•P
•B
• APO = ½ APB. Thus, APO = 42 Ans
10
TANGENT AND CHORD
EXAMPLES

Determine length EB in the figure
given below, given that AE = 4 in, CE
= 14 in, and ED = 2 in:
• (CE)(ED) = (AE)(EB)
C
E
A
D
B
14 in (2 in) = (4 in)(EB)
EB = 7 in Ans
11
ANGLES INSIDE A CIRCLE




An angle formed by two chords that intersect
within a circle is measured by one half the sum of
its two intercepted arcs
An inscribed angle is measured by one half its
intercepted arc
An angle formed by a tangent and a chord at the
tangent point is measured by one half its
intercepted arc
Find the length of arc AEB in the figure below,
given that CAB = 38:
C
•
A
•
+
E•
••
B
• CA is tangent to AB so CAB is
one half of arc AEB or, in other
words, arc AEB = 2CAB
Thus, arc AEB = 76° Ans
12
ANGLES OUTSIDE A CIRCLE


A
An angle formed outside a circle by two
secants, two tangents, or a secant and a
tangent is measured by one half the
difference of the intercepted arcs
Determine APD in the figure below,
given that arc AD = 98 and arc BC =
40:
• APD is equal to one half the
B
difference of arc AD and arc BC
P
C
APD = ½ (98 – 40) = 29 Ans
D
13
INTERNALLY AND EXTERNALLY
TANGENT CIRCLES



Two circles are internally tangent if both are
on the same side of the common tangent
line
Two circles are externally tangent if the
circles are on opposite sides of the common
tangent line
If two circles are either internally or
externally tangent, a line connecting the
centers of the circles passes through the
point of tangency and is perpendicular to the
tangent line
14
PRACTICE PROBLEMS
1.
Identify the central and inscribed angle in
the circle below:
A
B
D
2.
C
E
F
Define each of the following:
a. Tangent
b. Arc
c. Chord
15
PRACTICE PROBLEMS (Cont)
3.
4.
Determine the circumference of a
circle with a radius of 2.5 inches.
Determine the arc length of a circle
with a 5 m radius and a 50° arc.
16
PRACTICE PROBLEMS (Cont)
5.
Determine DB and arc ACB
in the figure at right, given
that AB = .6 m and arc AC
= .4 m.
B
+D
C
A
6.
Refer to the figure at right.
The circumference of the
circle is 110mm. Determine
these values:
6.
7.
The length of arc AB when 1
= 42°
1 when arc AB = 42 mm
A
+ 1
B
17
PRACTICE PROBLEMS (Cont)
7.
Find the value of ABC in the figure below,
given that arc AC = 100°.
A
B
C
8.
Find the number of degrees in arc DE in the
figure below, given that arc CF = 106° and
that EPD = 74.
E
D
C
P
F
18
PROBLEM ANSWER KEY
1.
2.
3.
4.
Central angle = ACB, inscribed angle =
EDF
a. A straight line that touches the circle
at only one point
b. That part of a circle between any two
points on the circle
c. A straight line segment that joins two
joints on the circle
15.71 inches
4.363 meters
19
PROBLEM ANSWER KEY (Cont)
5.
6.
7.
8.
a. DB = 0.3m
0.8m
a. 12.83mm
50°
42°
b. Arc ACB =
b. 137.45°
20