Circles> Formulas Assignment is Due

Center

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

Radius

Diameter

Chord

Tangent

Secant

Formulas

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

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

Center:

Radius:

Equation:

Center:

Radius:

Equation:

(-2,0)

4

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

(0,2)

2

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

16=(x+2)2+y2 4=x2+(y-2)2

Rewrite the equation of the circle in standard form and determine its

center and radius

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

(x+3)2 (y+5)2+ =22

Center: (-3,-5) Radius: 2

Rewrite the equation of the circle in standard form and determine its

center and radius

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

(x-7)2 (y+6)2+ =92

Center: (7,-6) Radius: 9

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.

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?

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

tangency.P

l

Q

If l is tangent to circle Q at P, then

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 32

10 = r

A16

24

r

r

B C

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.

1. 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

T

S

P If SR and TS are tangent to circle P, then

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

X2 – 7x+20 = 8

X2 7x+12= 0

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

X=3, x=4

B

D

C

AX2 -7x+20

8

Assignment

Angle Relationships

CentralInscribed

InsideOutside

Arc Length and Sector Area

n= arc measure

Find the length of Arc AB and the area of the shaded sector

Vocabulary:1. Minor Arc ________

2. Major Arc _______

3. Central Angle _______

4. Semicircle __________

DE

DBE

<DPE

BDP

B

D

E

Measure of Minor Arc = Measure of Central Angle

A

D

B

C

148

Find Each Arc:

a. CD_________

b. CDB ________

c. BCD _________

148

328

180

Measure of Minor Arc = Measure of Central Angle

Find Each Arc:

a. BD_________

b. BED ________

c. BE _________

142

218

118

A

E

B

C

D

100

6082

118

Inscribed Angle:

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

Inscribed Angle

Intercepted Arc

Example: Find the measure of the angleMeasure of Inscribed Angle = ½ the intercepted Arc

80

x

x = ½ the arc

x=1/2(80)

x=40

x

60

60 = ½ x

x=120

Find the measure of the ArcMeasure of Inscribed Angle = ½ the intercepted Arc

Example: Find the measure of each arc or angle

B

AC

D

mADC = ______180 mAC = _______

70

B

A

C

140

Find the measure of <BCA

m<BCA = ______36

B

AC

72

Find m<C

A

B

C

D

44

88

M<C = 44

Example:

Inside Angles– if two chords intersect in the interior 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

1

A

B

D

C

m<1 = ½( mDC + mAB)

Example: Find the missing angle

40

20AB

C

D

1

m<1 = ½( mDC + mAB)

m<1 = ½( 40+20)

m<1 = ½(60)

m<1 = 30

Outside Angles0 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.

1

A

B

C

m<1 = ½( mAB - mBC)

Example: find the missing angle

X = ½ (264-96)

X = ½ (168)

X=8496

X

264