Download Chapter 9 - 2015

Chapter 9
Circles
• Define a circle and a
sphere.
• Apply the theorems
that relate tangents,
chords and radii.
• Define and apply the
properties of central
angles and arcs.
9.1 Basic Terms
Objectives
• Define and apply the terms that describe a
circle.
The Circle
is a set of points in a plane at a given distance from a
given point in that plane
B
A
Blackout slide to show points
The Circle
The given distance from the center to any point on
the circle is a radius (plural radii)
How many radii
does a circle
have?
Are they all the
same length?
B
A
center
A
“Circle with center A”
Think – Pair - Share
•What is the
difference between
a line and a line
segment?
Chord
any segment whose endpoints are on the circle.
C
B
A
Diameter
A chord that contains the center of the circle
B
A
C
What is another
name for half of
the diameter?
Secant
any line that contains a chord of a circle.
C
B
A
Tangent
any line that contains exactly one point on the circle.
BC is tangent to
A
B
A
C
Point of Tangency
B
Point of tangency
A
Common Tangents
are lines tangent to more than one coplanar
circle.
Common External
Internal Tangents
Tangents
X
Y
B
B
R
A
X
X
B
D
Tangent Circles
are circles that are tangent to each other.
Internally Tangent
Circles
Externally
Tangent Circles
B
B
R
R
A
A
Sphere
is the set of all points in space equidistant from a
given point.
A
B
Sphere
Radii
Diameter
Chord
Secant
Tangent
C
E
B
A
F
D
Congruent Circles (or Spheres)
WHAT DO THEY HAVE?
• have equal radii.
B
E
A
D
Concentric Circles (or Spheres)
share the same center.
G
Who can think of a real world
example?
O
Q
Think , throwing of pointy
objects….
Inscribed/Circumscribed
A polygon is inscribed in a circle and the
circle is circumscribed about the polygon if
each vertex of the polygon lies on the circle.
L
P
N
M
Z
Name…
1. 3 radii
2. Diameter
3. Chord
R
4. Secant
5. Tangent
6. What is the name for ZX?
7. What should point M be called?
O
Q
X
White Boards….
• Draw the following…
1.
2.
3.
4.
5.
An inscribed triangle
A circle circumscribed about a quadrilateral
2 circles with common external tangents
2 circle that are internally tangent to each
other
9.2 Tangents
Objectives
• Apply the theorems that relate tangents and
radii
Experiment
• Supplies: Pencil, protractor, compass
1. Draw a circle with center A
2. Draw Point B on the bottom of your circle
3. Create line BC tangent to the circle at
Point B
4. Draw the radius to the point of tangency
and measure the angle formed by the
tangent and the radius (L ABC)
5. Compare your measurements with those
around you…
If the radius and tangent meet at the point of
tangency they form a right angle (perpendicular).
What can we
conclude based on
our experiment?
A
mABC  90
B
Point of
tangency
C
X
Dunce Cap Rule
Tangents to a circle from a
common point are
congruent.
Z
Y
How do we know the 2 right
triangles are congruent?
Inscribed/Circumscribed
When each side of a polygon is tangent to a
circle, the polygon is said to be circumscribed
about the circle and the circle is inscribed in
the polygon.
Each side of the poly, is
what to the circle?
• GIVEN
– Radius = 6
– BC = 8
– Find AC
• What allows you to
come up with the
correct answer?
A
B
• GIVEN
– LC
= 45
– Radius = 4
– Find AC and BC
BC is tangent to
C
A
Whiteboards
• Line XY is tangent to
D at X. The
radius of the circle is 7cm and the length of
segment XY is 4cm. FIND THE LENGTH
OF SEGMENT DY.
Whiteboards
• Create a diagram of the following…
1. A triangle circumscribed about a circle
2. A pentagon inscribed inside a circle
9.3 Arcs and Central Angles
Objectives
• Define and apply the properties of arcs and
central angles.
Experiment
Think – pair - share
How is the angle
measurement that you
just created related to
the measurement of a
circle?
Central Angle
Sides – 2 radii
Vertex – center of circle
L
Arc
an arc is part of a circle
like a segment is part
of a line.
C
A
B
AC
ABC
It’s like cutting
out a slice of
pizza!!
This represents
the crust of your
pizza
Central Angle / Arc Measure
the measure of an arc is given by the measure
of its central angle. (or vise versa)
80
A
C
80
m L ABC = 80
mAC  80
B
The central angle
tells us how much
of the 360 ◦ of
crust we are using
from our pizza.
Minor Arc
AC < 180
C
A
B
Semicircle
• “_____” a circle.
• What indicates that we have half a circle?
B
A
ABC
C
Major Arc
ACD > 180
B
A
C
D
We only know
how to
measure angles
up to 180, so
how do you
find the
measure of a
major arc?
Practice
Name two minor arcs
C
R
O
A
S
Practice
Name two major arcs
C
R
O
A
S
White Board Practice
Name two semicircles
C
R
O
A
S
Skip Remember….
• Adjacent angles ?
• Angle addition postulate?
•
Smartboard
Skip Adjacent Arcs
arcs that have exactly one point in common.
D
AD
A
C
B
DC
Arc Addition Postulate
D
A
B
mAD  mDC  mADC
C
Congruent Central L’s / Arcs
Theorem
In the same circle or in congruent circles….
• congruent arcs = congruent central angles
D
E
A
C
B
Dissecting a Circle Diagram
FREE VISUAL EVIDENCE !!!
• Central angles = minor arcs
• All the arcs = 360
• Diameters = straight lines = 180
• Vertical angles / adjacent angles
H
C

R
O
A
S
White Board Practice
H
Find the following
measurements…
C

30
RH
 AOR
HCA
100
50
210
R
O
50
A
S
Pg. 340
• #8 - 13
Section 9-4 Arcs and Chords
Objectives
• Define the relationships between arcs and
chords.
REVIEW
• WHAT IS A CHORD?
• WHAT IS AN ARC?
Relationship between a
chord and an arc
The minor arc between the endpoints of a
chord is called the arc of the chord, and the
chord between the endpoints of an arc is the
chord of the arc.
D
Chord BD “cuts off” 2 arcs..
BD and
BFD
B
A
F
Congruent Chords/Arcs Theorem
In the same circle or in congruent circles…
• congruent arcs have congruent chords
• congruent chords have congruent arcs.
D
If arc BD is congruent
to arc FC then…
C
B
A
F.
Skip - Midpoint/
Bisector of an Arc
• Just as we have learned about the bisectors and
midpoints of angles and line segments, an arc
can be bisected into two congruent arcs
D
If D is the midpoint of
arc BDC, then…….
C
B
A
Circle Handout Experiment
1. Label the center A
2. DRAW A CHORD AND LABEL IT DC
3. FIND THE MIDPOINT OF THE CHORD AND
LABEL IT X
4. DRAW A RADIUS THAT PASSES THROUGH
THE MIDPOINT AND INTERSECTS THE ARC
OF THE CIRCLE AT Y
5. USE A PROTRACTOR AND MEASURE LAXC
Think – Pair - Share
• What facts can you conclude about the arcs,
chords, or any other segments?
• What is congruent to what?
• What about perpendicular?
Chord Bisector Theorem
A radius or diameter perpendicular to a chord
bisects the chord and its arc.
D
DC  BY
Y
X
DX  XC
C
A
DY  YC
B
EC: What other 2 segments do I know are congruent that are not explicitly shown?
Remember
• When you measure distance from a point to
a line, you have to make a perpendicular
line.
A
Chord Distance Theorem
• In the same circle or in congruent circles:
• Chords equally distant from the center are congruent
• Congruent chords are equally distant from the center.
D
E
Y
B
X
A
C
Putting Pythagorean to Work..
Can we use the
given information
to make a
conclusion about
the chords shown?
(i.e. length of the
chords)
AY = 3 AX = 3
Radius = 4
D
E
Y
X
B
A
C
Hint: Just because
something is not
shown, doesn’t
mean it doesn’t
exist! (other radii)
THE HEAD BAND PROBLEM:
• Find the length of a chord 3cm from the
center of a circle with a radius of 7cm.
•
•
•
•
Workbook page 35
1a
2a
Rest on own
WHITEBOARDS
Solve for x and y
D
x
y
C
x = 12
y = 12
B
5
A
13
IF arc DB is 55 degrees, then arc CB is?
WHITEBOARDS
Solve for x and y
x=8
y = 16
8
y
x
WARM - UP
1. What does the term inscribed mean to
you in your own words?
–
Describe the placement of the vertices of an
inscribed triangle
2. What do we call the 2 sides and vertex (in circle
terms) of a central angle that you learned in
9.3?
–
What is the measure of the angle equal to?
9.5 Inscribed Angles
Objectives
• Solve problems and
prove statements
about inscribed
angles.
• Solve problems and
prove statements
about angles formed
by chords, secants
and tangents.
What we’ve learned…
• Inscribed means that something is inside of
something else – we have looked at inscribed
polys and circles
• We know that an angle by definition has a vertex
and 2 sides that meet at the vertex
• In a central angle…
– The vertex is the center and the sides are radii
Inscribed Angle
• Sides – two chords
• Vertex – point on the circle
A
“Intercepted arc”
C
What do you
think the sides
are in ‘circle
terms’? Where
is the vertex?
B
Experiment
1. Measure the inscribed angle created
with a protractor
2. Using the endpoints of the intercepted
arc, draw 2 radii to create a central
angle and then measure.
3. Compare the measurement of the
inscribed angle with that of the
central angle measure.
4. Discuss with your partner
Theorem
inscribed angle =
half the measure of the intercepted arc.
A
mABC  mAC
___
2
C
B
Corollary
If two inscribed angles intercept the same arc,
then they are congruent.
A
ABC  ADC
D
C
B
Don’t write down, just recgonize
WHITEBOARDS
• Find the values of r, s, x, y ,
and z
– Take inventory of the diagram before
trying to solve!
– Concentrate on parts of the whole
y◦
r◦
r = 50
s = 50
x = 160
y = 100
z = 100
x◦
O
s◦
80
z◦
Corollary
If the intercepted arc is a semicircle, the
inscribed angle must = 90.
A
What is the measure of
an inscribed angle
whose intercepted arc
has the endpoints of the
diameter?
B
O
mABC  90
C
Corollary
Quad inscribed = opposite angles supplementary.
mA  mC  180
mB  mD  180
A
B
O
C
D
Theorem
mABC  mADB
___
2
A
B
O
C
D
F
Treat this angle the same as you would an
inscribed angle!
Whiteboards
• Page 353
– #5, 6, 7
WHITEBOARDS
• Find the values of x, y , and z
60◦
X = 30
Y = 60
Z = 150
y◦
O
z◦
x◦
9.6 Other Angles
Objectives
• Solve problems and
prove statements
involving angles
formed by chords,
secants and
tangents.
WARM UP:
•For this diagram….
•Write down the different
equations that represent the
angle relationships shown.
•There’s more than one!
1
4
2
3
Partners: How do you think we can determine the measure of L1?
The angle formed by two intersecting chords
=
half the sum of the intercepted arcs.
B
m1  (mCB
 mDE )
_________
C
1
2
m2  (mCE
 mDB)
_________
2
2
E
A
D
Angles formed from a point
outside the circle…
2 secants
In each circle, 2
arcs are being
intercepted by the
angle.
2 tangents
1tangent
1 secant
The larger arc is
always further away
from the vertex.
Vertex outside the circle
=
half the difference of the intercepted arcs.
m1  (mBD
 mEF )
_________
B
2
E
C
s
m
a
l
l
1
F
l
a
r
g
e
A
D
WHITEBOARDS
• ONE PARTNER OPEN BOOK TO PG. 358
• ANSWER #1
– 35
• ANSWER # 6
– 40
• ANSWER # 4
– 80
• ANSWER # 7
– X=50
AB is tangent to circle O.
AF is a diameter
m AG = 100
B
m CE = 30
m EF = 25
8
A
C 6
D
30
E
25
3
7
O
2 1
100
G
5
4
F
WARM – UP
• READ PG. 361 – 363
– Identify what elements are involved in each of
the 3 theorems in this section
– Example: “Theorem 9-11 refers to the
relationship of 2 ___________ intersecting”
– What is the idea behind this section…. Angles,
segments, circles?
ANGLES QUIZ
•
Identify a numbered angle
that represents each of the
bullet points.
Write an equation
representing the measure of
the angle
•i.e. m L12 =
•
C
D
8
B
4
•X is center of circle
E
9
H
A
7
6
2
K
x
5
G
I
3
1
F
1.
2.
3.
4.
5.
Central angle
Inscribed angle
Angle formed inside
Angle formed outside
Name a 90 angle….