circles and arcs
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Chapter 10.6. Circles and arcs. Circle. A set of all points equidistant from the center. Center. Circle. A circle is named by the center. P. Circle P ( P). Diameter. A segment that contains the center of a circle and has both endpoints on the circle. Diameter. Radius. - PowerPoint PPT PresentationTRANSCRIPT
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CIRCLES AND ARCSChapter 10.6
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Circle A set of all points equidistant from the
center
Center
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Circle A circle is named by the center
Circle P (P)
P
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Diameter A segment that contains the center of a
circle and has both endpoints on the circle.
Diameter
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Radius A segment that has one endpoint at the
center of the circle and the other on the circle.
Radius
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Congruent Circles Congruent circles have the congruent
radii
P Q
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Central Angle An angle whose vertex is the center of
the circle.
Central Angle
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Arc Part of a circle. From point to point on
the outside of the circle.
Arc
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Semicircle An arc that’s half of the circle.
SemicircleHas a measure of 1800
1800
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Minor Arc A minor arc is smaller than half the
circle.
Minor Arc
400
Same measure as the corresponding interior angle
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Major Arc A major arc is larger than half the circle.
Major Arc360 minus the minor arc
400
3200
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Practice 1Name 3 of the following in A.1. the minor arcs2. the major arcs3. the semicircles
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Adjacent Arcs Adjacent arcs are arcs of the same circle that
have exactly one point in common.
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Arc Addition Postulate The measure of the arc formed by two adjacent
arcs is the sum of the measure of the two arcs.
400 7001100
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Practice 2 Find the measure of each arc in R.1. UT2. UV3. VUT4. ST5. VS
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Practice 3 Find each indicated measure for D.
1. mEDI2.3. mIDH4.
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Circumference The distance around the circle A measure of length
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Circumference The circumference of a circle is π times the
diameter (a = πd) or 2 times π and the radius (a = 2πr).
Diameter
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Circumference Example:
D = 4
C = d= 4
or = 12.52
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Circumference Example:
C = 2r= 2(5)
or = 31.4r = 5= 10
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Practice 4 Find the circumference of each circle.
Leave your answer in terms of .
1. 2.
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Arc Length The length of an arc is calculated using
the equation:
600
measure of the arc________________360 * circumference
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Arc Length The length of an arc is calculated using
the equation:
600
measure of the arc________________360 * d
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Arc Length The length of an arc is calculated using
the equation:
600
measure of the arc________________360 * 2r
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Arc Length________________measure of the arc
360 * d
600
7
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Arc Length________________ 60
360 * 7
600
7
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Arc Length________________ 1
6 * 22
600
7
= 3.67
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Practice 5 Find the length of each darkened arc.
Leave your answer in terms of .
1. 2.
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Area of a Circle The product of π and the square of the
radius.
A = r2
Radius
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Area of a Circle Example:
A = r2
= 52
or = 78.54r = 5= 25
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Practice 6 Find the area of a circle:
1. 6 in. radius
2. 10 cm. radius
3. 12 ft. diameter
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Sector of a Circle A sector of a circle is a region bounded
by an arc of the circle and the two radii to the arc’s endpoints.
You name a sector using the two endpoints with the center of the circle in the middle.
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Sector of a Circle Sector is the area of part of the circle
Area of blue section
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Area of Sector of a Circle The area of a sector is:
measure of the arc________________360 * r2
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Sector of a Circle Find the area of the sector
600
12
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Arc Length________________measure of the arc
360 * r2
600
12
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Arc Length________________ 60
360 * 122
600
12
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Arc Length________________ 1
6 * 144
600
12
= 24
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Segment of a Circle Part of a circle bounded by an arc and
the segment joining its endpoints
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Area of a Segment of a Circle Equal to the area of the sector minus the
area of a triangle who both use the center and the two endpoints of the segment.
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Sector – Triangle = Segment
Area of a Segment of a Circle
- =
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Area of a Segment of a Circle Find the area of the segment.
600
12
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Area of a Segment of a Circle Separate the triangle and the sector
600
12600
12
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Area of a Segment of a Circle Find the area of both figures
600
12600
12
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Area of Sector
600
12
________________ 60360 * 122
= 24
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Area of Triangle
600
6ð3
Find the altitude 12
or 10.4Find the base
6
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Area of Triangle
600
12
10.4
6
a = ½bh
= ½(12)(10.4)= 62.4
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Area of a Segment of a Circle Subtract the triangle from the Sector
24 62.4
-24 62.4 = 13