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10.3 – Apply Properties of Chords
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In the same circle, or in congruent circles, two ___________ arcs are congruent iff their corresponding __________ are congruent.
A
B C
D
AB CD
then
minorchords
ABª CDª
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If one chord is a _________________ _________ of another chord, then the first chord is the _________________.
PT PR
SQ
then
perpendicularbisector
diameter
andTR SQ
is the diameter
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If a ____________ of a circle is perpendicular to a chord, then the diameter ____________ the chord and its arc.
then
diameterbisects
EG DF
FH HD and FGª GDª
and the diameter
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In the same circle, or in congruent circles, two chords are congruent iff they are _________________ from the _____________.
then
equidistant center
EF AB
AB CD
and EG CD
EG EFand
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1. Find the given measure of the arc or chord. Explain your reasoning.
= 105°
Congruent chords
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1. Find the given measure of the arc or chord. Explain your reasoning.
= 360 4
Congruent chords
= 90°
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1. Find the given measure of the arc or chord. Explain your reasoning.
= 360 – 116 2
Congruent chords
= 122°
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1. Find the given measure of the arc or chord. Explain your reasoning.
=
Congruent arcs
6
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1. Find the given measure of the arc or chord. Explain your reasoning.
=
Diameter bisects chord
22
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1. Find the given measure of the arc or chord. Explain your reasoning.
= 119°
61°
119°Diameter bisects arc
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= 100°
50°50°
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360 – 85 – 65 2
=
= 105°
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Find the value of x.
3x + 16 = 12x + 7
16 = 9x + 7
9 = 9x
1 = x
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Find the value of x.
3x – 11 = x + 9
2x – 11 = 9
2x = 20
x = 10
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YES or NO
Reason:
_______________________ it is perpendicular and bisects
Tell whether is a diamter of . Explain.QS C
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Tell whether is a diamter of . Explain.QS C
YES or NO
Reason:
_______________________ it doesn’t bisect
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10.7 – Graphing Circles
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To come up with an equation of a circle, we need to express with an equation, the idea that its graph contains all the points that are equidistant from the center. If our center is at the origin, we would have a graph that looks like the following:
(x, y)r
r : ___________________________
x : ___________________________
y: ____________________________
Radius (distance from center)
Horizontal leg length of right
Vertical leg length of right
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Using Pythagorean theorem, we know that: ______________
For any equation of the form: ____________, the graph is the circle centered at the __________with a radius of r.
2 2 2 x y r
2 2 2 x y r
The circle must be then, the set of all points (x, y) that satisfy this equation.
origin
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1. Determine the radius of the circle whose equation is given:
2 2 16 x ya) 2 2 2 x y r
2 16r
r = 4
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b) 2 2 2 x y r
2 4r
2r
1. Determine the radius of the circle whose equation is given:
2 2 4 x y
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c) 2 2 2 x y r
2 5r
1. Determine the radius of the circle whose equation is given:
2 2 5 x y
5r
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2. Write the equation of a circle centered at the origin, whose radius is given:
9ra)
2 2 2 x y r
2 2 29 x y
2 2 81 x y
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7rb)
2 2 2 x y r
22 2 7 x y
2 2 7 x y
2. Write the equation of a circle centered at the origin, whose radius is given:
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2 5rc)
2 2 2 x y r
22 2 2 5 x y
2 2 20 x y
2. Write the equation of a circle centered at the origin, whose radius is given:
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We use horizontal and vertical shifts to move the center of the circle and get the standard form:
2 2 2 ( ) ( )
( , ) ____
x h y k r
whichhascenter and radius
h k r
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3. Find the center of the circle.
2 22 49 ( )x ya)
(0, –2)
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(–1, 7)
3. Find the center of the circle.
b) 2 21 7 9 ( ) ( )x y
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(3, 0)
3. Find the center of the circle.
c) 2 23 100 ( )x y
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4. Find the center, the radius, then graph the circle.
2 2 25x y a. Ctr ( , ) radius: r = __0 0 5
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b. Ctr ( , ) radius: r = __–4 0 32 2( 4) 9x y
4. Find the center, the radius, then graph the circle.
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c. Ctr ( , ) r = ______ 2 -1 72 2( 2) ( 1) 49x y
4. Find the center, the radius, then graph the circle.
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4. Find the center, the radius, then graph the circle.
d) Write the equation of the circle.
(x – 3)2 + (y – 1)2 = 4
Ctr ( , ) r = ______ 3 1 2
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5: Write the equation of a circle with center
(15, -9) and radius 4.2 2 2 x y r
2 2 215 9 4 x y
2 215 9 16 x y
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6: Write the equation of a circle with center (-4, 0) and radius 11.
2 2 2 x y r
2 224 11 x y
2 24 121 x y
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10.310.7
667-668692-695
3-9, 12-14Graphing Circles Worksheet
#8
8x – 13 = 6x + 9
2x – 13 = 92x = 22
x = 11