Transcript Chapter 1

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.
With your compass…
• Create a circle on your notes that is about
half the page
• As we go through the basic terms for a
circle label each one on your notes
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
The Circle
The given distance is a radius (plural radii)
How many radii
does a circle
have?
Are they all the
same length?
B
A
The Circle
The given point is the center
B
A
center
A
“Circle with center A”
The Circle
What are some real
world examples of
circles?
B
Point on circle
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 X
2. Draw Point Y on the bottom of your circle
3. Create line ZY tangent to the circle at
Point Y
4. Draw the radius to the point of tangency
and measure the angle formed by the
tangent and the radius (L XYZ)
5. Compare your measurements with those
around you…
Theorem
If a line is tangent to a circle, then the line is
perpendicular to the radius drawn to the
point of tangency.
B
What can we
conclude based on
our experiment?
A
C
mABC  90
Theorem (fill in the blank)
If a line in the plane of a circle is perpendicular
to a radius at its endpoint, then the line is
a tangent to the circle.
B
X
A
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
– BC = 4
– Find AC
BC is tangent to
C
A
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
1. Draw a straight line using a
straightedge
2. Extend the line so that its
sides go past the outside of
your protractor
3. Connect the two sides of
your angle using the outer
curved side of your
protractor
4. What do you end up with?
5. What do I get with a 70
degree angle?
Think – pair - share
How is the angle
measurement that you
just created related to
the measurement of a
circle?
Central Angle
is formed by two radii, with the center of the
circle as the vertex.
Arc
C
an arc is part of a circle
like a segment is part
of a line.
This represents
the crust of your
pizza
A
L
B
AC
It’s like cutting
out a slice of
pizza!!
ABC
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
an unbroken part of a circle with a measure
less than 180°.
C
AC
A
B
Semicircle
an unbroken part of a circle that shares endpoints
with a diameter.
How do I
know that
AC is a
diameter?
B
A
ABC
C
Major Arc
an unbroken part of a circle with a measure
greater than 180°.
ACD
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
AR, RC, RS, AS, SC
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
Theorem
In the same circle or in congruent circles….
• congruent arcs = congruent central angles
D
ABD  DBC
AD DC
A
C
B
smartboard
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
Group Practice
• Give the measure of each arc.
D
C
4x
E
2x
3x 3x + 10
A
B
Group Practice
m AB = 88
m BC = 52
m CD = 38
m DE = 104
m EA = 78
D
C
4x
E
2x
3x 3x + 10
A
B
The radius of the Earth is about 6400 km.
B
A
O
The latitude of the Arctic Circle is 66.6º
North. That means the m BE 66.6º.
B
A
66.6º
W
O
E
Find the radius of the Arctic Circle
xº
B
A
66.6º
W
O
E
Find the radius of the Arctic Circle
23.4º
B
A
66.6º
W
O
E
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
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?
Theorem
A diameter (or radius) that is 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
Putting Pythagorean to Work..
Partners: Use the
given information
do make a
conclusion about
the chords shown.
AY = 3 AX = 3
D
E
Y
X
B
A
4
C
Hint: Just because
something is not
shown, doesn’t
mean it doesn’t
exist! (other radii)
Theorem
• In the same circle or in congruent circles:
– Chords equally distant from the center (or centers) are
congruent
– Congruent chords are equally distant from the center (or
centers)
E
D
.
Y
YA  XA
BD  EC
B
X
A
C
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
Whiteboards
• Find the length of a chord 3cm from the
center of a circle with a radius of 7cm.
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
An angle formed by two chords or secant lines
whose vertex lies on the circle.
A
“Intercepted arc”
C
What do you
think the sides
are in ‘circle
terms’? Where
is the vertex?
B
Create a circle
using a compass
& an inscribed
angle in your
notes
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
The measure of an inscribed angle is 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
If a quadrilateral is inscribed in a circle, then
its opposite angles are supplementary.
mA  mC  180
mB  mD  180
A
B
O
C
D
Theorem
mABC  mADB
___
A
B
2
O
C
D
F
Treat this angle the same as you would and
inscribed angle!
Whiteboards
• Page 353
– #7, 6
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:
•Draw a central angle and label it
•Draw an inscribed angle and label it
•FOR BOTH CREATE AN EQUATION FOR THEIR
MEASURES IN COMPARISON TO THE
INTERCEPTED ARC
1
WARM UP:
•For this diagram….
•Write down the different
equations that represent the
angle relationships shown.
•There’s more than one!
4
2
3
Partners: How do you think we can determine the measure of L1?
Theorem
The angle formed by two intersecting chords
is equal to half the sum of the intercepted
arcs.
m1  (mCB
 mDE)
_________
C
1
2
m2  (mCE
 mDB)
_________
2
B
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.
Theorem
The angle formed by secants or tangents with the
vertex outside the circle has a measure equal to
half the difference of the intercepted arcs.
B
E
C
s
m
a
l
l
1
m1  (mBD
 mEF)
_________
2
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?
9.7 Circles and Lengths of
Segments
Objectives
• Solve problems about the lengths of chords,
secants and tangents.
Skip - Open Books to Pg. 361
• Read the paragraphs
for section 9-7
D
A
P
• “segments of a
chord”
C
O
– AP and PB
B
Theorem
If a chord intersects another
chord each chord now has
how many parts?
– If you multiply those 2
parts together they will
equal the product of the
other chords 2 parts.
B
E
X
F
D
EX  XD  FX  XB
Example
Solve for x
2 • 12 = 3 • x
24 = 3x
8
= x
2
x
3
12
Skip - Pg. 362 P
• Read the middle
paragraph
B
D
• “external segments”
– aka “outside piece”
– BP and DP
O
• “Secant segments”
– Aka “whole piece”
– AP and CP
C
A
Theorem
2 Secants
•Each one is made up of an outside part and an inside part that make
the whole.
B
E
C
CB  CE  CD  CF
F
A
D
Whole  Outside Part = Whole  Outside Part
Example
Whole  Outside Part = Whole  Outside Part
8 •x = 12 •4
8x = 48
x =6
8
x
4
8
Skip Pg. 362 and 363
• Read the bottom of
362 and top of 363
P
B
A
• “external segments”
– BP and PD
• “Secant segments”
D
– AP and PD
Whole  Outside Part = Whole  Outside Part
Theorem
1 Secant
•Still the same parts… outside + inside = whole
1 Tangent
•Only made up of an outside part…which is also the
whole!!
E
C
CD  CF  CE  CE
F
A
D
Whole  Outside Part = Whole  Outside Part
Example
Whole Piece Outside Piece = Whole Piece Outside Piece
x •x = 24 •6
x2 = 144
x = 12
6
x
18
WHITEBOARDS
• ONE PARTNER OPEN BOOK TO PG. 363/364
• ANSWER #1
–x=8
• ANSWER # 4
–x=4
• ANSWER # 5
–x=4
WHITEBOARDS
5
x
4
3
y
X =9
Y =6
ANGLES QUIZ
C
•
D
8
B
•
4
Identify a numbered angle
that represents each of the
bullet points.
Write an equation
representing the measure
of the angle
•i.e. m L12 =
E
H
A
7
6
2
x
K
5
3
1
G
I
F
1.
2.
3.
4.
5.
Central angle
Inscribed angle
Angle formed inside
Angle formed outside
90 angle