Download Name: Chapter 10 Notes Lesson 10-1 Center: the middle Radius: A

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Chapter 10 Notes
Lesson 10-1
Center:
the middle
Radius:
A line segment drawn from the center to any point on the circle
Diameter:
A line segment drawn across a circle that passes through the center
(twice the radius)
Chord:
Any segment with endpoints that are on the circle
Circumference:
The distance around the outside of the circle; C = 2πr, C = πd
Lesson 10.2 Angles and Arcs
• The sum of the measures of the central angles of a circle with no interior points is
360.
• Minor arc- arc whose central angle is less than 180
• Major arc- arc whose central angle is greater than 180
• Semicircle- half a circle, 180
• Arc addition postulate- the measure of the arc formed by 2 adjacent arcs is the
sum of the measures of the 2 arcs.
A
l

360 2r
Example 1: Refer to circle T
a. Find mRTS
b. Find mQTR
Your Turn: Refer to circle Z.
a. Find mCZD
b. Find mBZC
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Example #2:
Use circle P where mNPM  46, PL bisects KPM , and OP  KN.
a. Find mOK .
b. Find mLM .
c. Find mJKO
Your Turn:
In circle B, XP and YN are diameters, mXBN  108,
and BZ bisects YBP. Find each measure.
a. mYZ
b. mXY
c. mXNZ
Example #3: BICYCLES This graph shows the percent of each type of bicycle sold in the
United States in 2001.
a. Find the measurement of the central angle representing
each category. List them from least to greatest.
b. Is the arc for the wedge named Youth congruent to the arc for
the combined wedges named Other and Comfort?
Your Turn: SPEED LIMITS This graph shows the percent of U.S. states that have each
speed limit on their interstate highways.
a. Find the measurement of the central angles representing
each category. List them from least to greatest.
b. Is the arc for the wedge for 65 mph congruent to the
combined arcs for the wedges for 55 mph and 70 mph?
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Example #4: In circle B, AC = 9 and mABD  40 . Find the length of AD.
Your Turn: In circle A, AY = 21 and mXAY  45 . Find the length of WX.
Lesson 10.3 Arcs and Chords
Theorem 10.2- In a circle or in congruent circles, two minor arcs are congruent if and only
if their corresponding chords are congruent
Theorem 10.3- In a circle, if a diameter (or radius) is perpendicular to a chord, then it
bisects the chord and its arc.
Theorem 10.4- In a circle or in congruent circles, two chords are congruent if and only if
they are equidistant from the center.
Example #2: TESSELLATIONS The rotations of a tessellation can create twelve
congruent central angles. Determine whether PQ  ST . .
Your Turn: ADVERTISING A logo for an advertising campaign is a pentagon that has five
congruent central angles. Determine whether PTS  PAS.
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Example 3: Circle W has a radius of 10 centimeters. Radius WL is perpendicular to chord
HK which is 16 centimeters long.
a. If mHL = 53, find mMK.
b. Find JL.
Your Turn: Circle O has a radius of 25 units. Radius OC is perpendicular to the chord AE ,
which is 40 units long.
a. If mMG = 35, find mCG.
b. Find CH.
Example #4: Chords EF and GH are equidistant from the center.
If the radius of circle P is 15 and EF = 24, find PR and RH.
Your Turn: Chords SZ and UV are equidistant from the center of circle X.
If TX = 39 and XY = 15, find WZ and UV.
Lesson 10.4 Inscribed Angles
 Theorem 10.5 Inscribed Angle Theorem
If an angle is inscribed in a circle, then the measure of the angle equals one-half the
measure of its intercepted arc.
 Theorem 10.6
If two inscribed angles of a circle (or congruent circles) intercept congruent arcs or the
same arc, then the angles are congruent
 Theorem 10.7
If an inscribed angle intercepts a semicircle, the angle is a right angle
 Theorem 10.8
If a quadrilateral is inscribed in a circle, then its opposite angles are supplementary.
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Example #1: In circle F, mWX = 20, mXY = 40, mUZ = 108, and mUW = mYZ.
Find the measures of the numbered angles.
Your Turn: In circle A, mXY = 60, mYZ = 80 and mWX = mWZ.
Find the measures of the numbered angles.
Example #3: PROBABILITY Points M and N are on a circle so that MN = 72. Suppose
point L is randomly located on the same circle so that it does not coincide with M or N.
What is the probability that mMLN  144 ?
Your Turn: PROBABILITY Points A and X are on a circle so that mAX = 84. Suppose point
B is randomly located on the same circle so that it does not coincide with A or X. What is
the probability that mABX  42 ?
Example #4: ALGEBRA Triangles TVU and TSU are inscribed in circle P
with VU = SU. Find the measure of each numbered angle if
m2  x  9 and m4  2x  6 .
Your Turn: ALGEBRA Triangles MNO and MPO are inscribed in circle D with MN = OP. Find
the measure of each numbered angle if m2  4x  8 and m3  3x  9
Example #5: Quadrilateral QRST is inscribed in circle M. If mQ  87 and mR  102
Find mS and mT .
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Your Turn: Quadrilateral BCDE is inscribed in circle X. If mB  99 and mC  76
Find mD and mE .
Lesson 10.5 Tangents
• Theorem 10.9
If a line is tangent to a circle, then it is perpendicular to the radius drawn at the point of
tangency
• Theorem 10.10
If a line is perpendicular to the radius of a circle at its endpoint on the circle, then the
line is tangent to the circle.
• Theorem 10.11
If two segments from the same exterior point are tangent to a circle, then they are
congruent.
Example #1: Algebra RS is tangent to circle Q at point R. Find y.
Your Turn: CD is a tangent to circle B at point D. Find a.
Example #2a:
Determine whether BC is tangent to circle A.
to circle D.
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Example 2b:
Determine whether WE is tangent
Your Turn:
Determine whether ED is tangent to circle Q.
to circle V.
Determine whether XW is tangent
Example #3: ALGEBRA Find x. Assume that segments that appear tangent to circles are
tangent.
Your Turn: ALGEBRA Find a. Assume that segments that appear tangent to circles are
tangent.
Example #4: Triangle HJK is circumscribed about circle G.
Find the perimeter of HJK if NK  JL  29
Your Turn: Triangle NOT is circumscribed about circle M.
Find the perimeter of NOT if CT  NC  28
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Lesson 10.6 Secants, Tangents, and Angle Measures
• Theorem 10.12
If two secants intersect in the interior of a circle, then the measure of an angle formed is
one-half the sum of its measure of the arcs intercepted by the angle and its vertical angle.
• Theorem 10.13
If a secant and tangent intersect at a point of tangency, then the measure of each angle
formed is one-half the measure of the intercepted arc.
• Theorem 10.14
If two secants, a secant and a tangent, or two tangents intersect in the exterior of a
circle, then the measure of the angle formed is one-half the positive difference of the
measures of the intercepted arcs.
Ex.#1: Find m4 if mFG  88 and mEH  76.
Your Turn: Find m5 if mAC  63 and mXY  21.
Example #2: Find RPS if mPT = 114 and mTS = 136.
Your Turn: Find FEG if mHF = 80 and mHE = 164.
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Example #3: Find x.
Your Turn: Find x.
Example #4: JEWELRY A jeweler wants to craft a pendant with the shape shown.
Use the figure to determine the measure of the arc at the bottom of the pendant.
Your Turn: PARKS Two sides of a fence to be built around a circular garden in a park are
shown. Use the figure to determine the measure of A.
Example #5: Find x.
Your Turn: Find x.
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Lesson 10.7
• Theorem 10.5
If two chords intersect in a circle, then the products of the measures of the segments of
the chords are equal.
• Theorem 10.16
If two secant segments are drawn to a circle from an exterior point, then the product of
the measures of one secant segment and its external secant segment is equal to the
product of the measures of the other secant segment and its external secant segment.
• Theorem 10.17
If a tangent segment and a secant segment are drawn to a circle from an exterior point,
then the square of the measure of the tangent segment is equal to the product of the
measures of the secant segment and its external secant segment.
Example #1: Find x.
Your Turn: Find x.
Example #2: BIOLOGY Biologists often examine organisms under microscopes. The circle
represents the field of view under the microscope with a
diameter of 2 mm. Determine the length of the organism
if it is located 0.25 mm from the bottom of the field of
view. Round to the nearest hundredth.
Your Turn: ARCHITECTURE
Phil is installing a new window in an addition for a client’s home.
The window is a rectangle with an arched top called an eyebrow.
The diagram below shows the dimensions of the window.
What is the radius of the circle containing the arc if the
eyebrow portion of the window is not a semicircle?
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Example #3: Find x if EF = 10, EH = 8, and FG = 24.
Your Turn: Find x if GO = 27, OM = 25, and IK = 24.
Example #4: Find x. Assume that segments that appear to be tangent are tangent.
Your Turn: Find x.
Assume that segments that appear to be tangent are tangent.
Lesson 10.8
An equation for a circle with a center at (h, k) and a radius of r( xunits
 h) 2 is:
 ( y  k )2  r 2
Example #1a: Write an equation for a circle with the center at (3, –3), d = 12.
Example #1b: Write an equation for a circle with the center at (–12, –1), r = 8.
Your Turn: Write an equation for each circle.
a. center at (0, –5), d = 18
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b. center at (7, 0), r =20
Example #2: A circle with a diameter of 10 has its center in the first quadrant. The lines y
= –3 and x = –1 are tangent to the circle. Write an equation of the circle.
Your Turn: A circle with a diameter of 8 has its center in the second quadrant. The lines y
= –1 and x = 1 are tangent to the circle. Write an equation of the circle.
Example #3a: Graph  x  2   ( y  3) 2  4
y
2
Example #3b:  x  3  y 2  16
2
y
x
x
y
Your Turn:
a. Graph x 2  ( y  5) 2  25
y
b. Graph  x  4  ( y  3) 2  9
2
x
Example #4: ELECTRICITY Strategically located substations are extremely important in
the transmission and distribution of a power company’s electric supply. Suppose three
substations are modeled by the points D(3, 6), E(–1, 0), and F(3, –4). Determine the
location of a town equidistant from all three substations, and write an equation for the
circle.
Your Turn: AMUSEMENT PARKS The designer of an amusement park wants to place a
food court equidistant from the roller coaster located at (4, 1), the Ferris wheel located
at (0, 1), and the boat ride located at (4, –3). Determine the location for the food court
and write an equation for the circle.
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x