10.3 – apply properties of chords. in the same circle, or in congruent circles, two ___________...

10.3 – Apply Properties of Chords

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

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

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

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

1. Find the given measure of the arc or chord. Explain your reasoning.

= 105°

Congruent chords

1. Find the given measure of the arc or chord. Explain your reasoning.

= 360 4

Congruent chords

= 90°

1. Find the given measure of the arc or chord. Explain your reasoning.

= 360 – 116 2

Congruent chords

= 122°

1. Find the given measure of the arc or chord. Explain your reasoning.

=

Congruent arcs

6

1. Find the given measure of the arc or chord. Explain your reasoning.

=

Diameter bisects chord

22

1. Find the given measure of the arc or chord. Explain your reasoning.

= 119°

61°

119°Diameter bisects arc

= 100°

50°50°

360 – 85 – 65 2

=

= 105°

Find the value of x.

3x + 16 = 12x + 7

16 = 9x + 7

9 = 9x

1 = x

Find the value of x.

3x – 11 = x + 9

2x – 11 = 9

2x = 20

x = 10

YES or NO

Reason:

_______________________ it is perpendicular and bisects

Tell whether is a diamter of . Explain.QS C

Tell whether is a diamter of . Explain.QS C

YES or NO

Reason:

_______________________ it doesn’t bisect

10.7 – Graphing Circles

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

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

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

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

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

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

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:

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:

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

3. Find the center of the circle.

2 22 49 ( )x ya)

(0, –2)

(–1, 7)

3. Find the center of the circle.

b) 2 21 7 9 ( ) ( )x y

(3, 0)

3. Find the center of the circle.

c) 2 23 100 ( )x y

4. Find the center, the radius, then graph the circle.

2 2 25x y a. Ctr ( , ) radius: r = __0 0 5

b. Ctr ( , ) radius: r = __–4 0 32 2( 4) 9x y

4. Find the center, the radius, then graph the circle.

c. Ctr ( , ) r = ______ 2 -1 72 2( 2) ( 1) 49x y

4. Find the center, the radius, then graph the circle.

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

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

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

10.310.7

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3-9, 12-14Graphing Circles Worksheet

#8

8x – 13 = 6x + 9

2x – 13 = 92x = 22

x = 11