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TRANSCRIPT
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WARM UP
10987654321
Sunday, November 16, 2014
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Objectives
• Use properties of circles to derive the formula for sector area.
• Determine arc length and arc measure for given central and inscribed angle
measures
• Determine angle measures using the properties of interior angles.
HomeworkCircle Packet, Sections I, II and III all problems
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ALL Make Up Tests for the Log and Exponents Unit must be
completed by Monday November 17th .
No exceptions.
ALL Retakes for the Log and Exponents Unit must be
completed by Friday November 21st.
No exceptions.
You MUST bring your test corrections with you to be eligible for a retake.
UNIT TEST THIS FRIDAY
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Homework review: Sections III, IV and V
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Homework review
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Homework review
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Homework review
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Homework review
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Homework review
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Homework review
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Homework review
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Homework review
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Homework review
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Homework review: Sections VI
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Homework review: Mid-segment Theorem
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Homework review: Triangles Section 6
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Circles: Just what are they good for?
Apparently these people found a good use for them!
We will find even better uses for circles.
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The GUTS of a circle…
Radius
Circumference
Diameter
Area
𝑪 = 𝟐𝝅𝒓
𝑨 = 𝝅𝒓𝟐
Divides the circle into two equal parts.
Straight line connecting the center to the edge of the circle.
Half the length of the diameter.
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Radius
Every radius in the same circle is
equal to all the other radii in the
same circle.
So what is the value of x?
X
5X-32
𝟓𝒙 − 𝟑𝟐 = 𝒙𝒙 = 𝟖
The GUTS of a circle…
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Chord
Line drawn inside the circle that
touches the circle at its endpoints.
Chords are named by their end
points.D
C
B
AThe GUTS of a circle…
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Circles guts
Central Angle
Vertex is the center of the circle and
it extends to the edge of the circle.
C
A B
∠𝑨𝑩𝑪 is a central angle
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Circles guts
Inscribed Angle
Vertex is on the edge of the circle
and extends inward to the opposite
edge of the circle.
F
D
E∠𝑫𝑬𝑭 is an inscribed angle
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Quick Check
What is the circumference of a circle?
WRONG
Pie are round!
Did you say “pi are squared”?
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Circles guts
Arc
Part of the circumference cut off by
an angle.
C
A B
The measure of an Arcs is based on
the central angle that intercepts the
Arc.
𝒓 = 𝟓
𝟒𝟓°
𝑚 𝑎𝑐 = 45
An Arc can be referred to by its
measure or its length.
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Circles guts
Arc
C
A B
The length of an Arc is based on its
portion of the circumference.
𝒓 = 𝟓
𝟒𝟓°
𝑙 =1
82𝜋𝑟
What percentage of this circle’s
circumference does this arc represent?
𝑝𝑒𝑟𝑐𝑒𝑛𝑡 =𝑝𝑎𝑟𝑡
𝑤ℎ𝑜𝑙𝑒=45
360=1
8
The arc length is equal to this percentage
time the circumference of the circle.=1
82𝜋(5)=
5
4𝜋
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Circles guts
Sectors
A sector is like a pizza slice. It is a
portion of the area of a circle. 90°
𝑠 =1
4𝜋𝑟2
What percentage of this circle’s area does
this sector represent?
𝑝𝑒𝑟𝑐𝑒𝑛𝑡 =𝑝𝑎𝑟𝑡
𝑤ℎ𝑜𝑙𝑒=90
360=1
4
The sector area is equal to this percentage
time the area of the circle.=1
4𝜋(52) =
25
4𝜋
𝒓 = 𝟓
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Central Angles vs. Inscribed Angles
A central angle is equal to the
measure of its intercepted arc.
𝒎∠𝑨𝑶𝑩 = 𝒎 𝑨𝑩 = 𝟖𝟎°
An inscribed angle is equal to one
half the measure of its intercepted
arc.
𝒎∠𝑨𝑩𝑪 =𝟏
𝟐𝒎 𝑨𝑪 = 𝟓𝟎°
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1. 2. 3.
Find X (no pointing)
𝑥 = 113 𝑥 = 62 𝑥 = 192
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Use the properties we just discussed to find the measure of angle ABC.
What do you notice about angle ABC
and angle ADC?
They intercept the same arc!
How does that help you determine the
value of x?
Because they intercept the same arc the
two angles have the same measure.
Therefore the measure of angle ABC is also 81o.
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More Inside Angles.
But first a word about
tangents.
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There is a special relationship between tangents and the arcs they intercept.
Notice these angles are inside the circle!
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Write this formula down. It’s not in your packet. (It’s in your study guide)
Tangent Chord Angle =1
2Intercepted arc
∠𝑀𝐿𝐾 =1
2𝑚 𝐾𝐿
=1
2(130)
= 65
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Find the measure of ∠𝑆𝑅𝑄
73
Find the measure
of major arc 𝐺𝐹
208
You try these two…
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=1
2(170 + 70)
Lets talk about angles formed by two intersecting chords.
We can use the intercepted arcs to find
angle measures.
Angle formed inside by intersecting chords
= 1
2Sum of intersected arcs
∠𝐵𝐸𝐷 =1
2(𝑚 𝐴𝐶 +𝑚 𝐵𝐷)
= 120
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∠𝐵𝐷𝐶 =1
2(𝑚 𝐵𝐶 +𝑚 𝐴𝐸)
Sometimes we have to work backward
Find 𝑚 𝐵𝐶
85 =1
2(𝑚 𝐵𝐶 + 70)
170 = 𝑚 𝐵𝐶 + 70
100 = 𝑚 𝐵𝐶
Use the formula
Fill in what you know
Solve for what
you don’t know
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You try this one
∠𝑈𝑆𝑇 =1
2(𝑚 𝑈𝑇 +𝑚 𝑉𝐴)
Find m 𝑈𝑇
120 =1
2(𝑚 𝑈𝑇 + 60)
240 = 𝑚 𝑈𝑇 + 60
180 = 𝑚 𝑈𝑇
Use the formula
Fill in what you know
Solve for what
you don’t know
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To Summarize
Inscribed
Angle
Tangent
Angle
Intersecting
Chord
Angles
Central
Angle
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PERFECT PRACTICE MAKES PERFECT!
Work on your Circles packet.
If you finish it today’s assignment before you leave, you
can accumulate credits for a 2 point addition to your unit
test grade.
You’ll need 5 credits for 2 points.
Problems 24-31 in your study guide