circles

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Circles
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This photograph was taken 216 miles above Earth.
From this altitude, it is easy to see
the curvature of the horizon. Facts about circles
can help us understand details about Earth.
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circle all points that are the same distance from
the center of the circle. A circle with center C
is called circle C, or ?C.
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Definitions

• A radius is a line segment from the center to
  the circle.
• A diameter is a chord that passes through the
  center.

radius
center
diameter
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• The word radius (plural radii) is also used to
  denote the length of a radius (all radii have the
  same length).
• The word diameter is also used to denote the
  length of a diameter (all diameters have the same
  length).
• Note that the diameter of a circle is twice its
  radius.

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All circles are similar to each other. Congruent
circles have the same size radius
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• 1. If DE is a tangent line,
• What is the angle x?
• Because DE is a tangent line
• It must be perpendicular to the radius/diameter
  at the point of tangency.
• Angle D must be a right angle
• 180 38 90 52

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• 2. If AB is tangent to circle C,
• Find the radius.
• Tangent is perpendicular
• Use Pythagorean Theorem
• C2 B2 A2
• (x8)2 122 x2
• x2 16x 64 144 x2
• 16x 80
• x 5

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• Is ML tangent to Circle N
• at L? Explain.
• If ML is tangent, than it will
• create a right angle at the
• point of tangency, and right triangle NLM
• We can use the Pythagorean theorem to check if it
  is a right triangle
• 72 242 252
• 49 576 625
• 625 625 , ?NLM is a right triangle
• ?L is 90 degrees
• LM is perpendicular to the radius at L, so
• LM is tangent to Circle N

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• LM and MN are both
• tangents to circle O.
• Find x . Hint Sum of
• internal angles in a quadrilateral is 360
• Since LM and MN are tangents, they are
  perpendicular to the radius, creating 90 angles.
  
• 90 117 90 x 360
• 297 x 360
• x 63

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Example 3 Problem Solving Application
Early in its flight, the Apollo 11 spacecraft
orbited Earth at an altitude of 120 miles. What
was the distance from the spacecraft to Earths
horizon rounded to the nearest mile? Assume
earths radius is 4000 miles
The answer will be the length of an imaginary
segment from the spacecraft to Earths horizon.
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EC CD ED
Seg. Add. Post.
Substitute 4000 for CD and 120 for ED.
4000 120 4120 mi
EC2 EH² CH2
Pyth. Thm.
Substitute the given values.
41202 EH2 40002
974,400 EH2
Subtract 40002 from both sides.
987 mi ? EH
Take the square root of both sides.
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The problem asks for the distance to the nearest
mile. Check if your answer is reasonable by using
the Pythagorean Theorem. Is 9872 40002 ? 41202?
Yes, 16,974,169 ? 16,974,400.
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Check It Out! Example 3
Kilimanjaro, the tallest mountain in Africa, is
19,340 ft tall. What is the distance from the
summit of Kilimanjaro to the horizon to the
nearest mile? (5,280 ft in a mile)
The answer will be the length of an imaginary
segment from the summit of Kilimanjaro to the
Earths horizon.
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Given
ED 19,340
Change ft to mi.
EC CD ED
Seg. Add. Post.
4000 3.66 4003.66mi
Substitute 4000 for CD and 3.66 for ED.
EC2 EH2 CH2
Pyth. Thm.
Substitute the given values.
4003.662 EH2 40002
Subtract 40002 from both sides.
29,293 EH2
Take the square root of both sides.
171 ? EH
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The problem asks for the distance from the summit
of Kilimanjaro to the horizon to the nearest
mile. Check if your answer is reasonable by using
the Pythagorean Theorem. Is 1712 40002 ? 40042?

Yes, 16,029,241 ? 16,032,016.
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• Two Tangents from the same point create congruent
  segments from that point to the points of
  tangency.

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• 1. Circle O is inscribed in triangle
• ABC. What is the triangles
• Perimeter?
• Since AD and AF both come from
• The same point A and are tangent, AD?AF
• We can find each of the missing segments this way
• Perimeter AD BD BE CE CF AF
• Perimeter 10 15 15 8 8 10
• Perimeter 66

10cm
15cm
8cm
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• 2. Circle O is inscribed in triangle
• ABC which has a perimeter of
• 88cm. What is the length of
• QY?
• The inscribed circle has points of tangency
  at X, Y and Z.
• PX?PZ and RZ ? RY and QY ? QX
• Perimeter PX PZ ZR RY QY QX
• 88 15 15 17 17 QY QY
• 88 64 2QY
• 24 2QY, QY 12, QX 12

15cm
17cm
Substitute QY for QX since they are equal
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A central angle vertex is the center of a circle.
An arc is an unbroken part of a circle
consisting of two points called the endpoints and
all the points on the circle between them.
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Minor arcs may be named by two points. Major arcs
and semicircles must be named by three points.
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• Find the radius of circle K
• KN ? LM given
• KN bisects LM
• LM 14
• LN 7
• Use Pythagorean Theorem
• r2 32 72
• r2 58
• r 7.6cm

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• 2. Find the value for y
• BC ? AF Given
• Draw an auxiliary line BA
• Creating right triangle ABC
• AB 15, it is a radius
• Use Pythagorean Theorem
• 152 y2 112
• 104 y2
• 10.2 y

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Definitions
inscribed angle an angle whose vertex is on a
circle and whose sides contain chords of the
circle. intercepted arc consists of endpoints
that lie on the sides of an inscribed angle and
all the points of the circle between them. A
chord or arc subtends an angle if its endpoints
lie on the sides of the angle.
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Example 1 Finding Measures of Arcs and Inscribed
Angles
Find each measure.
m?PRU
Inscribed ? Thm.
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Example 1B Finding Measures of Arcs and
Inscribed Angles
Find each measure.
Inscribed ? Thm.
Substitute 27 for m? SRP.
Multiply both sides by 2.
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Example 2 Finding Angle Measures in Inscribed
Triangles
Find a.
?WZY is a right angle ?WZY is inscribed in a semicircle.
m?WZY 90? Def of rt. ?
5a 20 90 Substitute 5a 20 for m?WZY.
5a 70 Subtract 20 from both sides.
a 14 Divide both sides by 5.
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The area of a sector is a fraction of the circle
containing the sector. To find the area of a
sector whose central angle measures m, multiply
the area of the circle by
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Example 1A Finding the Area of a Sector
Find the area of each sector. Give answers in
terms of ? and rounded to the nearest hundredth.
sector HGJ
Use formula for area of sector.
Substitute 12 for r and 131 for m.
Simplify.
52.4? m2 ? 164.62 m2
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Example 1B Finding the Area of a Sector
Find the area of each sector. Give answers in
terms of ? and rounded to the nearest hundredth.
sector ABC
Use formula for area of sector.
Substitute 5 for r and 25 for m.
Simplify.
? 1.74? ft2 ? 5.45 ft2
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Check It Out! Example 1a
Find the area of each sector. Give your answer in
terms of ? and rounded to the nearest hundredth.
sector ACB
Use formula for area of sector.
Substitute 1 for r and 90 for m.
Simplify.
0.25? m2 ? 0.79 m2
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Check It Out! Example 1b
Find the area of each sector. Give your answer in
terms of ? and rounded to the nearest hundredth.
sector JKL
Use formula for area of sector.
Substitute 16 for r and 36 for m.
Simplify.
25.6? in2 ? 80.42 in2
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Example 2 Automobile Application
A windshield wiper blade is 18 inches long. To
the nearest square inch, what is the area covered
by the blade as it rotates through an angle of
122?
Use formula for area of sector.
r 18 in.
? 345 in2
Simplify.
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Check It Out! Example 2
To the nearest square foot, what is the area
watered in Example 2 (p. 765) as the sprinkler
rotates through a semicircle?
Use formula for area of sector.
d 720 ft, r 360 ft
? 203,575 ft2
Simplify.
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In the same way that the area of a sector is a
fraction of the area of the circle, the length of
an arc is a fraction of the circumference of the
circle.
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Example 4A Finding Arc Length
Find each arc length. Give answers in terms of ?
and rounded to the nearest hundredth.
FG
Use formula for area of sector.
Substitute 8 for r and 134 for m.
? 5.96? cm ? 18.71 cm
Simplify.
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Example 4B Finding Arc Length
Find each arc length. Give answers in terms of ?
and rounded to the nearest hundredth.
an arc with measure 62? in a circle with radius 2
m
Use formula for area of sector.
Substitute 2 for r and 62 for m.
? 0.69? m ? 2.16 m
Simplify.
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Check It Out! Example 4a
Find each arc length. Give your answer in terms
of ? and rounded to the nearest hundredth.
GH
Use formula for area of sector.
Substitute 6 for r and 40 for m.
Simplify.
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Check It Out! Example 4b
Find each arc length. Give your answer in terms
of ? and rounded to the nearest hundredth.
an arc with measure 135 in a circle with radius
4 cm
Use formula for area of sector.
Substitute 4 for r and 135 for m.
3? cm ? 9.42 cm
Simplify.