Download Note Sheets Chapter 6: Discovering and Proving Circle Properties

Ch 6 Note Sheets L2 Shortened Key
Name ___________________________
Note Sheets Chapter 6:
Discovering and Proving Circle Properties
Review: Circles Vocabulary
If you are having problems recalling the vocabulary, look back at your notes for Lesson 1.7 and/or page 69 – 71 of your book.
Also, pay close attention to the geometry notations you need to use to name the parts!!
Circles
G
Circle is a set of points in a plane a given distance
(radius) from a given point (center).
F
R
Congruent circles are two or more circles with the
same radius measure.
Concentric circles are two or more circles with the
same center point.
A
C
P
Types of Lines [Segments]
Radius [of a circle] is
• A segment that goes from the center to any point
on the circle.
• the distance from the center to any point on the
circle.
• All radii of a circle are equal.
Diameter [of a circle] is
• A chord that goes through the center of a circle.
• Diameter is the longest chord in a circle.
• the length of the diameter. d = 2r or ½ d = r.
Chord is a segment connecting any two points on the
circle.
Secant A line that intersects a circle in two points.
Tangent [to a circle] is a line that intersects a circle in
only one point. The point of intersection is called the
point of tangency.
Arcs & Angles
Arc [of a circle] is formed by two points on a circle and a
continuous part of the circle between them. The two
points are called endpoints.
Semicircle is an arc whose endpoints are the endpoints of
the diameter.
Minor arc is an arc that is smaller than a semicircle.
Major arc is an arc that is larger than a semicircle.
Intercepted Arc An arc that lies in the interior of an
angle with endpoints on the sides of the angle.
Central angle An angle whose vertex lies on the center
of a circle and whose sides are radii of the circle.
Measure of a central angle determines the measure of
an intercepted minor arc.
S. Stirling
B
D
E
radii: AP , PR , PB
diameter: AB
chords: AB , CD , AF
secant: AG or AG
tangent: EB
semicircles: ACB , ARB
minor arcs: AC , AR , RD , BC ,…
major arcs: ARC , ARD , FRD ,
BAC ,…
central angles with their intercepted
arcs:
m∠APR = mAR
m∠RPB = mRB
Page 1 of 8
Ch 6 Note Sheets L2 Shortened Key
Lesson 6.1 Tangent Properties
Investigation 1: See Worksheet page 1.
Tangent Conjecture
A tangent to a circle is
perpendicular to
the radius drawn to the
point of tangency.
Name ___________________________
Converse of the Tangent
Conjecture
A line that is perpendicular
to a radius at its endpoint
on the circle is tangent to
the circle.
O
T
N
Investigation 2: See Worksheet page 1.
Tangent Segments Conjecture
Tangent segments to a circle from a point outside
the circle are congruent.
A
E
G
Intersecting Tangents Conjecture
The measure of an angle formed by two intersecting tangents
to a circle is 180 minus the measure of the intercepted arc,
x = 180 − a .
x
a
a
b
“Tangent” means to intersect in one point.
Tangent Circles Circles that are tangent to the same line at the same point.
Internally Tangent
Circles
Two tangent circles
having centers on the
same side of their
common tangent.
Externally Tangent
Circles
Two tangent circles
having centers on
opposite sides of their
common tangent.
Polygons and Circles
The triangle is inscribed in the circle, or
the circle is circumscribed about the triangle.
The triangle is circumscribed about the circle, or
the circle is inscribed in the triangle.
B
All of the vertices
of the polygon are
on the circle.
The sides are all
chords of the circle.
P
C
All of the sides of the polygon
are tangent to the circle.
O
C
A
A
S. Stirling
B
Page 2 of 8
Ch 6 Note Sheets L2 Shortened Key
Example 1
Name ___________________________
Example 2
Rays r as s are tangents. w = ?
180 – 54 = w
OR
Tangent ⊥ radius, so m∠ONM = m∠OPM = 90
Quad. sum = 360 so 360 − 180 − 110 = x , x = 70
y = 5 cm Tangents to a circle from a point are congruent.
mNP = 110 Measure of a central angle equals its intercepted
arc.
so central angle = w
now use the kite formed by =
tangent segments and = radii
360 – 90 – 90 – 54 = w
126 = w
mPQN = 360 − 110 = 250 Total degrees in a circle = 360°
Example 3
AD is tangent to both circle B and circle C.
w = 100 Quadrilater sum = 360
mAXT = 260 Circle’s degree = 360
m∠A = 90 and m∠D = 90 Tangent ⊥ radius, w = 360 − 90 − 90 − 80 = 100
w = mAT = 100 Measure of central angle = intercepted arc. mAXT = 360 − 100 = 260
S. Stirling
Page 3 of 8
Ch 6 Note Sheets L2 Shortened Key
Lesson 6.2 Chord Properties
Read top of page 317. Then use the
diagrams to define the following.
Name ___________________________
Examples:
Non-Examples:
R
D
P
Central angle
An angle whose vertex lies on the
center of a circle and whose sides
are radii of the circle.
Measure of a central angle =
measure of its intercepted arc.
Inscribed angle
An angle whose vertex lies on a
circle and whose sides are chords of
the circle.
O
Q
A
S
T
B
∠PQR , ∠PQS , ∠RST ,
∠QST and ∠QSR are NOT
∠AOB , ∠DOA and
∠DOB are central
angles of circle O.
central angles of circle O.
Examples:
Non-Examples:
Q
A
C
V
R
P
B
E
T
W
X
D
∠ABC , ∠BCD and
∠CDE are inscribed
U
S
∠PQR , ∠STU , and ∠VWX
are NOT inscribed angles.
angles.
Investigation 3: See Worksheet page 4.
Example 1
Chord Conjectures
If two chords in a circle are congruent, then
• they determine two central angles that
are congruent.
• their intercepted arcs are congruent.
S. Stirling
B
D
If AB ≅ CD , then
O
∠BOA ≅ ∠COD .
If AB ≅ CD , then
A
AB ≅ CD
Page 4 of 8
C
Ch 6 Note Sheets L2 Shortened Key
Lesson 6.3 Arcs and Angles
Read top of page 324.
Review (from Chapter 1.7):
A circle measures 360°.
Name ___________________________
C
Example 1
If m∠COR = 92° ,
then m∠CAR = ? .
92
O
A
R
A semicircle measures 180°.
mCR = 92° .
The central angle equals its intercepted arc.
Investigation 4: See Worksheet page 6.
If mCR = 92° , then m∠CAR = 46° .
The inscribed angle equals half its
intercepted arc.
Inscribed Angle Conjecture
The measure of an angle inscribed in a circle
equals half the measure of its intercepted arc.
Example 2
mAB = 170° , find m∠APB and m∠AQB .
A
P
m∠APB = 1 (170 ) ° = 85°
2
m∠AQB = 1 (170 ) ° = 85°
2
170
Q
An inscribed angle measures ½ the intercepted arc.
B
Example 3
m AB = mCD = 42° , find m∠APB and m∠CQD
m∠APB = 1 i42 = 21°
2
1
m∠CQD = i42 = 21°
2
The inscribed angle equals half its intercepted arc.
m∠ACD = m∠ABD = m∠AED = 90°
.
S. Stirling
B
O
42
Q
D
R
A
Find the measure of each angle.
2
42
C
Example 4
∠ACD , ∠ABD and ∠AED intercept semicircle ARD .
All are 1 i180 = 90°
A
P
O
C
B
Page 5 of 8
D
E
Ch 6 Note Sheets L2 Shortened Key
Name ___________________________
Lesson 6.5 The Circumference/Diameter Ratio Read top of page 335.
Pi is a number, just like 3 is a number. It represents the number you get when you take the circumference of a
circle and divide it by the diameter. There is no exact value for pi, so you use the symbol π. You will leave π in the
answers to your problems unless they ask for an approximate answer. If they do, use 3.14 or the π key on your
calculator.
Circumference
The perimeter of a circle, which is the distance around the circle. Also, the curved path of
the circle itself.
Circumference Conjecture
If C is the circumference and d is the diameter of a circle, then there is a number π such
that C = dπ
Since d = 2r, where r is the radius, then C = 2rπ or 2πr
Example A: If a circle has a diameter of 3 meters,
what is its circumference?
Example B: If a circle has a circumference
of 12π meters, what is its radius?
C =πd
C = π •3
C = 3π
C ≈ 9.4 m
C = 2π r
12π = 2π r
12π 2π r
=
2π
2π
r = 6m
write the formula
substitute
simplify (pi written last, like a variable)
Only an approximate answer!
write the formula
substitute
divide to get r.
simplify
Example C:
If a circle has a circumference of 20 meters, what
is its diameter?
C =πd
20 = π d
20
=d
π
write the formula
substitute
accurate answer
d ≈ 6.366 m approximate
S. Stirling
Page 6 of 8
Ch 6 Note Sheets L2 Shortened Key
Lesson 6.7 Arc Length
Investigation 8: See Worksheet page 11.
Name ___________________________
Measure of an Arc: The measure of an arc equals the measure of its central angle,
measured in degrees.
Arc length: The portion of (or fraction of) the circumference of the circle described by an
arc, measured in units of length.
Arc Length Conjecture
The length of an arc equals the measure of the arc divided by 360° times the
circumference.
It is a fraction of the circle! So…….
Use the formula every time!! Arc Length =
Example (a):
Given central ∠ATB = 90° with radius 12.
B
mAB = 90°
fraction of circle =
length of AB =
=
90 1
=
360 4
arc degrees
• Circumference
360
Example (b):
Given central ∠COD = 180° with diameter 15
cm.
90
90
T
mCED = 180°
A
90
( 2π i12)
360
1
( 24π ) = 6π
4
D
180 1
=
360 2
180
length of CED =
(15π )
360
15
= π cm
2
fraction of circle =
Example (c):
Given mEF = 120° with radius = 9 ft.
fraction of circle =
120 1
=
360 3
1
( 2π i9)
3
= 6π ft
length of EF =
S. Stirling
120
F
O
E
Page 7 of 8
O
C
Ch 6 Note Sheets L2 Shortened Key
Name ___________________________
EXAMPLE B:
If the radius of a circle is 24 cm and
EXAMPLE C:
The length of ROT is 116π, what is the radius of
the circle?
m∠BTA = 60° , what is the length of AB ?
mAB = 120°
B
120
( 2π i24)
360
= 16π ≈ 50.3 cm
O
120
length of AB =
60
T
S. Stirling
A
mROT = 360 − 120° = 240
240
116π =
( 2π ir )
360
4π
116π =
r
3
116π 3
i =r
then
1 4π
r = 87
R
120
Page 8 of 8
T