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Circles. > Formulas Assignment is Due. Tangent. Center. Radius. Diameter. Chord. Secant. Circle: A circle is the set of all points in a plane that are equidistant from a given point called the center of the circle. A circle with center P is called “ circle P ” or P. Formulas.

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circles

Circles

> Formulas Assignment is Due

slide2

Tangent

Center

Radius

Diameter

Chord

Secant

Circle: A circle is the set of all points in a plane that are equidistant from a given point called the center of the circle. A circle with center P is called “circle P” or P

standard equation of a circle
Standard Equation of a Circle

r2 = (x-h)2 + (y-k)2

Where,

r = radius

(h,K) = center of the circle

example write the standard equation of a circle with center 2 1 and radius 2
Example: Write the standard equation of a circle with center (2,-1) and radius = 2

r2 = (x-h)2 + (y-k)2

22 = (x- 2)2 + (y- -1)2

4 = (x-2)2 + (y+1)2

example give the coordinates for the center the radius and the equation of the circle
Example: Give the coordinates for the center, the radius and the equation of the circle

(0,2)

Center:

Radius:

Equation:

Center:

Radius:

Equation:

(-2,0)

2

4

22=(x-0)2+(y-2)2

42=(x-(-2))2+(y-0)2

4=x2+(y-2)2

16=(x+2)2+y2

rewrite the equation of the circle in standard form and determine its center and radius
Rewrite the equation of the circle in standard form and determine its center and radius

x2+6x+9+y2+10y+25=4

+

=22

(x+3)2

(y+5)2

Center: (-3,-5)

Radius: 2

rewrite the equation of the circle in standard form and determine its center and radius1
Rewrite the equation of the circle in standard form and determine its center and radius

x2-14x+49+y2+12y+36=81

+

=92

(x-7)2

(y+6)2

Center: (7,-6)

Radius: 9

slide9
Use the given equations of a circle and a line to determine whether the line is a tangent or a secant

Circle: (x-4)2 + (y-3)2 = 9

Line: y=-3x+6

example the diagram shows the layout of the streets on mexcaltitlan island
Example: The diagram shows the layout of the streets on Mexcaltitlan Island.

1. Name 2 secants

2. Name two chords

3. Is the diameter of the circle greater than HC?

4. If ΔLJK were drawn, one of its sides would be tangent to the circle. Which side is it?

slide11

P

l

Q

THM: If a line is tangent to a circle, then it is perpendicular to the radius drawn to the point of tangency.

If l is tangent to circle Q at P, then

if bc is tangent to circle a find the radius of the circle

r

A

16

r

B

C

24

If BC is tangent to circle A, find the radius of the circle.

Use the pyth. Thm.

r2+242 = (r+16)2

r2+576 = (r+16)(r+16)

r2+576 = r2+16r+16r+256

r2+576 = r2+32r+256

-r2 -r2

  • 576 = 32r + 256
  • -256 -256
  • 320 = 32r
  • 32
  • 10 = r
slide13

Example: A green on a golf course is in the shape of a circle. A golf ball is 8 feet from the edge of the green and 28 feet from a point of tangency on the green, as shown at the right. Assume that the green is flat.

  • What is the radius of the green

2. How far is the golf ball from the cup at the center?

thm if 2 segments from the same exterior point are tangent to a circle then they are congruent

R

S

P

If SR and TS are tangent to circle P, then

T

Thm: If 2 segments from the same exterior point are tangent to a circle, then they are congruent.
ab and da are tangent to circle c solve for x

X2-7x+20

B

A

C

8

D

AB and DA are tangent to circle C. Solve for x.

X2 – 7x+20 = 8

X27x+12= 0

(x-3)(x-4)=0

X=3, x=4

angle relationships

Angle Relationships

Central

Inscribed

Inside

Outside

vocabulary

DE

DBE

BD

Vocabulary:
  • Minor Arc ________
  • Major Arc _______
  • Central Angle _______
  • Semicircle __________

<DPE

measure of minor arc measure of central angle

Find Each Arc:

  • CD_________
  • CDB ________
  • BCD _________
Measure of Minor Arc = Measure of Central Angle

148

328

180

measure of minor arc measure of central angle1

Find Each Arc:

  • BD_________
  • BED ________
  • BE _________
Measure of Minor Arc = Measure of Central Angle

118

142

218

118

inscribed angle
Inscribed Angle:

An angle whose vertex is on a circle and whose sides contain chords of the circle.

Intercepted Arc

Inscribed Angle

example find the measure of the angle

80

x

Example: Find the measure of the angle

Measure of Inscribed Angle = ½ the intercepted Arc

x= ½ the arc

x=1/2(80)

x=40

find the measure of the arc

x

60

Find the measure of the Arc

Measure of Inscribed Angle = ½ the intercepted Arc

60 = ½ x

x=120

example find the measure of each arc or angle

B

70

B

A

C

C

A

D

mADC = ______

mAC = _______

Example: Find the measure of each arc or angle

140

180

find the measure of bca

B

72

C

A

Find the measure of <BCA

m<BCA = ______

36

find m c

B

44

A

C

D

Find m<C

M<C = 44

88

inside angles

D

m<1 = ½( mDC + mAB)

C

1

A

B

Inside Angles

– if two chords intersect in theinterior of a circle, then the measure of each angle is one half the sum of the measures of the arcs intercepted by the angle and its vertical angle

example find the missing angle

20

A

B

1

C

m<1 = ½( mDC + mAB)

40

D

Example: Find the missing angle

m<1 = ½( 40+20)

m<1 = ½(60)

m<1 = 30

outside angles

A

m<1 = ½( mAB - mBC)

C

1

B

Outside Angles
  • If a tangent and a secant, two tangents, or two secants intersect in the exterior of a circle, then the measure of the angle formed is one half the difference of the measures of the intercepted arcs.
example find the missing angle1

96

X

Example: find the missing angle

264

X = ½ (264-96)

X = ½ (168)

X=84