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Geometry 2205
Unit 4:
Mrs. Bondi
Geometry
Unit 4: Circles
Unit 4 Circles Topics:
Lesson 1:
Lesson 2:
Lesson 3:
Lesson 4:
Lesson 5:
Lesson 6:
Lesson 7:
Lesson 8:
Circles and Arcs (PH text 10.6)
Areas of Circles, Sectors, and Segments of Circles (PH text 10.7)
Geometric Probability (PH text 10.8)
Tangent Lines (PH text 12.1)
Chords and Arcs (PH text 12.2)
Inscribed Angles (PH text 12.3)
Angle Measures and Segment Lengths (PH text 12.4)
Circles in the Coordinate Plane (PH text 12.5)
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Geometry 2205
Unit 4:
Mrs. Bondi
Lesson 1: Circles and Arcs (PH text 10.6)
Objectives:
to find the measures of central angles and arcs of circles.
to find the circumference of a circle and the length of an arc
Circle: the set of all points equidistant from a given point called the ______________
Circles are named using the symbol “
” and the center point.
Radius:
Diameter:
Congruent Circles:
Central Angle:
The center P of the circle is the midpoint of the diameter! Remember the midpoint formula:
Example 1: A diameter of a circle has endpoints A(-3, -2) and B(1, 4). Find the coordinates of the
center and find the length of the radius.
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Geometry 2205
Unit 4:
Mrs. Bondi
Arc:
Arc notation:
Three Types of Arcs:
Semicircle:
Minor Arc:
Major Arc:
Named by:
Named by:
Named by:
Measure:
Measure:
Measure:
Example:
Find each arc in the diagram to the right and its measure.
Adjacent Arcs:
Postulate 10-2
Arc Addition Postulate: The measure of the arc formed
by two adjacent arcs is the sum of the measure of the two arcs.
mABC  mAB  mBC
Example 2:
L
35̊
P
165̊
15̊
M
N
Find the measure of each arc:
a. MN
b. LM
d. LKN
c. KLM
K
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Geometry 2205
Unit 4:
Mrs. Bondi
Circumference -
 = pi =
Theorem 10-9
Circumference of a Circle
The circumference of a circle is  times the diameter.
C
d
= ratio of circumference
to diameter of a circle
C   d or C  2 r
Example 3:
The diameter of a circle is 16 cm. Find the circumference. (round to tenth)
Example 4:
The radius of a circle is 4 in. Find the circumference. (round to tenth)
Concentric Circles –
Example 5:
Compare circumferences
d = 12
r = 4 cm
c
m
Arc Length –
Theorem 10-10
Arc Length
The length of an arc of a circle is the product of the ratio
measure of the arc
360
and the circumference of the circle.
O
length of AB 
r
mAB
 2 r
360
B
A
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Geometry 2205
Unit 4:
Mrs. Bondi
Example 6:
If a  60 , diameter = 12, find length AB .
(round to nearest tenth)
O
a
B
A
Example 7: C
B
If radius = 12 cm, ACB = 210, find length ACB .
(round to nearest tenth)
A
Congruent Arcs -
Practice:
HW: p.654 #8-34 even
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Geometry 2205
Unit 4:
Mrs. Bondi
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Geometry 2205
Unit 4:
Mrs. Bondi
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Geometry 2205
Unit 4:
Mrs. Bondi
Lesson 2: Areas of Circles, Sectors, and Segments of Circles (PH text 10.7)
Objective:
to find the areas of circles, sectors and segments of circles
Theorem 10-11
Area of a Circle The area of a circle is the product
of  and the square of the radius.
A   r2
Example 1:
The radius of a circle is 15 cm. Find the area.
A
Sector of a Circle – region bounded by two radii and their intercepted arc.
B
r
Theorem 10-12
Area of a Sector of a Circle
The area of a sector of a circle is the product of the ratio
measure of the arc
360
and the area of the circle.
Area of sector AOB 
Example 2: The diameter of a circle is 8.2 m,
and m AB = 125. Find the area of sector ADB.
Round to the nearest tenth.
Example 3:
Find the area of sector GPH
Leave your answer in terms of
B
C
mAB
 r 2
360
D
A
8
.
Geometry 2205
Unit 4:
Mrs. Bondi
Segment of a circle – The part of a circle bounded by an arc and the
segment joining its endpoints.
Example 4: A circle has radius 8 cm. Find the area of a segment of the circle bounded by a 120
degree arc. Round your answer to the nearest tenth.
120̊
8 cm
Example 5: What is the area of the shaded segment shown at the right? Round your answer to the
nearest tenth.
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Geometry 2205
Unit 4:
Mrs. Bondi
Practice:
HW: p.663 #6-30 even
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Geometry 2205
Unit 4:
Mrs. Bondi
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Geometry 2205
Unit 4:
Mrs. Bondi
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Geometry 2205
Unit 4:
Mrs. Bondi
Lesson 3: Geometric Probability (PH text 10.8)
Objective:
to use segment and area models to find the probability of events
Geometric probability uses geometric figures to model occurrences of real-life events. The
occurrences can be compared by comparing the measurements of the figures.
Reminder:
Probability is …
P(event) = # of favorable outcomes
# of possible outcomes
Examples:
1)
2.
3. KYW gives a weather update every 10 minutes. If you turn on the radio at a random time, what is
the probability that you would wait more than 4 minutes to hear the weather update?
P(wait > 4 min.) =
P(wait ≤ 2 min.) =
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Geometry 2205
Unit 4:
Mrs. Bondi
4. Bill takes the train to work in Center City each morning. Because of traffic, he cannot be sure
exactly when he will arrive at the station. If trains leave the station every 20 minutes during rush
hour, what is the probability he will not need to wait more than 5 minutes for a train to leave?
Examples:
5.
6.
7. What is the probability of a tossed coin landing in the shaded region of the rectangle below?
The rectangle measures 125 cm long and 25 cm wide.
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Geometry 2205
Unit 4:
Mrs. Bondi
8. Rebekah and Bridget made a dodecagon shaped dartboard for a school carnival. It is divided into a
series of triangles by connecting opposite vertices alternately colored red, orange, yellow, green, blue and
purple.
a) What is the probability that the dart will land in a yellow triangle?
b) What is the probability that the dart will land in a red or orange triangle?
c) What is the probability that the dart will land in a primary color triangle?
d) What is the probability that the dart will land in a triangle whose color name contains the letter r?
Practice:
HW: p.671 #8-40 even
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Geometry 2205
Unit 4:
Mrs. Bondi
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Geometry 2205
Unit 4:
Mrs. Bondi
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Geometry 2205
Unit 4:
Mrs. Bondi
Lesson 4: Tangent Lines (PH text 12.1)
Objective:
to use properties of a tangent to a circle
A
B
Point of Tangency
Tangent to a circle
A Tangent to a Circle Point of Tangency –
Theorem 12-1
If a line is tangent to a circle, then it is perpendicular to
the radius drawn to the point of tangency.
If AB is tangent to
Example 1:
O at P, then AB  OP .
Example 2:
Example 3:
What is the distance to the horizon that a person can see on a clear day from an airplane 2 miles above
the earth? The Earth’s radius is about 4000 mi.
Theorem 12-2
(Converse of 12-1)
If a line in the same plane as a circle is perpendicular to a radius at
its endpoint on the circle, then the line is tangent to the circle.
If AB  OP at P, then AB is tangent to
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O.
Geometry 2205
Unit 4:
Mrs. Bondi
Example 4:
Find the length of the radius if
PQ = 8 and TQ = 5.
P
8
Example 5:
Find the length of the segment connecting the
centers of the circles.
20 in
Q
5
T
6 in
10 in
S
Theorem 12-3
If two tangent segments to a circle share a common endpoint
outside the circle, then the two segments are congruent.
If AB and CB are tangent to
respectively, then AB  CB .
O at A and C,
The sides of the triangle to the right are tangent to the circle.
The circle is “inscribed in” the triangle.
The triangle is “circumscribed about” the circle.
O is inscribed within ∆PQR. Label each length that you know based on Thm 12-4.
Example 5:
Q
15 cm X
P
O
Y
If ∆PQR has a perimeter of 88cm, find QY.
Z
17 cm
R
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Geometry 2205
Unit 4:
Mrs. Bondi
Practice:
HW: p.766 #5-19
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Geometry 2205
Unit 4:
Mrs. Bondi
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Geometry 2205
Unit 4:
Mrs. Bondi
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Geometry 2205
Unit 4:
Mrs. Bondi
Lesson 5: Chords and Arcs (PH text 12.2)
Objective:
to use congruent chord, arcs and central angles
to use perpendicular bisectors to chords
P
Chord – a segment with endpoints on a circle
– labeled as a line segment, PQ
O
– marks an arc on the circle, labeled PQ
Q
Theorem 12-4 and its converse
In the same circle or in congruent circles,
a) congruent central angles have congruent arcs
b) congruent arcs have congruent central angles
Theorem 12-5 and its converse
In the same circle or in congruent circles,
a) congruent central angles have congruent chords
b) congruent chords have congruent central angles
Theorem 12-6 and its converse
In the same circle or in congruent circles,
a) congruent chords have congruent arcs
b) congruent arcs have congruent chords
B
D
Let’s prove Theorem 12-6 part a.
Given:
O
P
BC  DF
Statements
O
Prove: BC  DF
Reasons
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P
C
F
Geometry 2205
Unit 4:
Mrs. Bondi
Theorem 12-7 and its converse
In the same circle or in congruent circles,
c) chords equidistant from the center(s) are congruent
d) congruent chords are equidistant from the center(s)
Example 1:
answer.
Find the value of x in the diagram to the right. Justify your
Theorem 12-8
In a circle, if a diameter is perpendicular to a
chord, then it bisects the chord and its arc.
Theorem 12-9
In a circle, if a diameter bisects a chord (that is not a
diameter), then it is perpendicular to the chord.
Theorem 12-10
In a circle, the perpendicular bisector of a chord
contains the center of the circle.
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Geometry 2205
Unit 4:
Mrs. Bondi
Example 2:
Let chord CE  30 in. and 6 in. from the center.
A
6 in
T
30 in
C
E
A
a)
Find the radius of
Example 3:
a)
A
b)
Find mCE
T
C
Let chord AB  22 in. and 7 in. from the center.
Find the radius of
C
C
A
b)
E
D
B
Find the length of AB
C
A
Practice:
25
D
B
Geometry 2205
Unit 4:
Mrs. Bondi
3. What is the missing length to the nearest tenth?
HW: p.776 #5-15, 30-32
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Geometry 2205
Unit 4:
Mrs. Bondi
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Geometry 2205
Unit 4:
Mrs. Bondi
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Geometry 2205
Unit 4:
Mrs. Bondi
Lesson 6: Inscribed Angles (PH text 12.3)
Objective:
to find the measure of an inscribed angle
to find the measure of an angle formed by a tangent and a chord
A
C  inscribed angle
AB  intercepted arc of C
O
B
ABC inscribed in
O
C
Theorem 12-10
Inscribed Angle Theorem
The measure of an inscribed angle is half the measure of its intercepted arc.
mB 
1
m AC
2
Example 1:
a
P
mPQT  70
mTS  25
T
70 
Find a and b
25 
Q
S
b
R
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Geometry 2205
Unit 4:
Mrs. Bondi
Example 2:
A
B
F
1)
Name a pair of congruent inscribed angles
2)
Name a right angle
3)
Name a pair of supplementary angles
C
E
D
Example 3:
107
Find the value of each variable.
z
x y
98 o
73
Example 4:
Find the value of each variable.
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Geometry 2205
Unit 4:
Mrs. Bondi
Theorem 12-12
The measure of an angle formed by a chord and a tangent (that
intersect on a circle) is half the measure of the intercepted arc.
A
Example 5:
Find the value of each variable.
y
x
C
z
58 
B
Practice:
D
HW: p.784 #6-18, 20-21, 23-25
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Geometry 2205
Unit 4:
Mrs. Bondi
Extra Practice:
Good for Review: Mid-Chapter quiz p.788
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Unit 4:
Mrs. Bondi
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Geometry 2205
Unit 4:
Mrs. Bondi
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Geometry 2205
Unit 4:
Mrs. Bondi
Lesson 7: Angle Measures and Segment Lengths (PH text 12.4)
Objective:
to find the measures of angles formed by chords, secants and tangents.
to find the lengths of segments associated with circles
Secant – a line (or segment, or ray) that intersects a circle at two points
– a secant always contains a chord
Theorem 12-13
The measure of an angle formed by two lines (or chords)
that intersect inside a circle is half the sum of the measures of the intercepted arcs.
m1 
1
 x  y
2
Theorem 12-14
The measure of an angle formed by
two lines that intersect outside a
circle is half the difference of the
measures of the intercepted arcs.
m1 
1
 x  y
2
Examples: Find the value of the variables.
55 
1.
2.
35 
103
3.
4.
130
90
170
x
y
35 
39 
y
35
x
Geometry 2205
Unit 4:
Mrs. Bondi
B
5.
6.
z
72 
A
x
55 
80 
C
D
Theorem 12-15 Summary:
Case I:
The products of the chord segments are equal
Case II:
The products of the secants and their outer segments are equal
Case III: The product of a secant and its outer segment equals the square of the tangent
Examples:
7.
8.
16
x
x
7
9
12
11
5
9.
10. What is the length of AB ?
15
10
12
z
A
B
3
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Geometry 2205
Unit 4:
Mrs. Bondi
11.
Find the value of the variables.
2
y
3
11
6
x
8
Practice:
HW: p.794 #8-20, 24-26
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Geometry 2205
Unit 4:
Mrs. Bondi
Extra Practice:
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Geometry 2205
Unit 4:
Mrs. Bondi
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Geometry 2205
Unit 4:
Mrs. Bondi
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Unit 4:
Mrs. Bondi
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Geometry 2205
Unit 4:
Mrs. Bondi
Lesson 8: Circles in the Coordinate Plane (PH text 12.5)
Objective:
to write the equation of a circle
to find the center and radius of a circle
Think about some points that would make this equation true.
x 2  y 2  25
Imagine a compass connecting these points in a circular shape.
Theorem 12-16
The standard form of an equation of a circle with center (h, k) and radius r is
x  h 2   y  k 2  r 2
This is easily derived from the distance formula.
d
 x2  x1 
Examples:
2

 y 2  y1

2
Find the center and radius.
1.
x2 + y2 = 9
2.
x2 + y2 = 36
3.
x2 + y2 = 121
4.
(x – 6)2 + (y – 2)2 = 25
5.
(x – 8)2 + (y + 4)2 = 225
6.
(x + 3)2 + (y + 7)2 = 100
Example 7: Write the standard equation of each circle.
a) center (3, -4) and radius 6
b)
42
center (-2, -1) and radius
2
Geometry 2205
Unit 4:
Mrs. Bondi
Using Algebra:
When given the center and a point on the circle, …
The distance formula can be used to find the circle’s radius.
The midpoint of the diameter is the center of the circle.
What is the standard equation of the circle with center (1, -3)
that passes through the point (2, 2).
distance formula to find radius:
Use radius (computed) and center (given) to write an equation.
Example 8:
A diameter of a circle has endpoints (-3, 7) and (5, 5). Write an equation of the circle.
Practice:
HW: p.801 #8-38 even
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Geometry 2205
Unit 4:
Mrs. Bondi
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Geometry 2205
Unit 4:
Mrs. Bondi
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Geometry 2205
Unit 4:
Mrs. Bondi
Circle Relationships
Visualize at a glance the relationships between the various angles, arcs, and sectors of a circle.
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