4Circles.notebook

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November20,2013

O

DONOW:1.Whatarecoordinatesofthecenterofthecircle?

2.Whatisthelengthoftheradius?

CIRCLES

(h,k)r

The equation of a circle:

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

where (h,k) represents the CENTER

and r represents the RADIUS

To find the center and radius when you are given the

equation of a circle:

-switch the signs for the coordinates of the center

-take the square root of "r2" to get the radius

For each of the following circles, find the center and the radius:

1) x2 + y2 = 25 Center: __________ Radius: ___________

2) (x - 4)2 + (y + 8)2 = 4 Center: __________ Radius: ____________

3) (x + 9)2 + (y - 1)2 = 10 Center: __________ Radius: ____________

4) (x - 2)2 + y2 = 5 Center: __________ Radius: ____________

Given the following, write an equation for each circle.

5) Center: (-3,2) Radius: 6

6) Center: (0, -8) Radius: 1

7) Center: (-7,-10)Radius: 9

8) Write an equation of a circle whose center is

(3, -5) and whose diameter is 10.

Writing the Equation of a Circle Given a Graph:

9) Write an equation for circle O shown below.

10) Write an equation for the circle shown below.

Circles: VOCABULARY

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Example3

Example5

APandBPtouchthecircle___________

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EXAMPLES1) Given: CE = 12, ED = 12, and AE = 3, find BE.

2) In the diagram of circle O below, chord AB intersects

chord CD at E, DE = 2x + 8, EC = 3, AE = 4x - 3, and

EB = 4. What is the value of x?

3) Chords AB and CD intersect at point E in a circle with

center at 0. If AE = 8, AB = 20, and DE = 16, what is the

length of CE? EXTERNAL times WHOLE secant = EXTERNAL times WHOLE secant

E D

C

AB

ew=ew

External segment of a secant --> part of secant segment that is

outside the circle.

Intersecting Secants

4) In the accompanying diagram, PAB and PCD are

secants drawn to circle O, PA = 8, PB = 20, and PD = 16.

What is the length of PC?

5) Secants JKL and JMN are drawn to circle O from an

external point, J. If JK = 8, LK = 4, and JM = 6, what is

the length of JN?

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6) In the diagram below of circle O, secant AB intersects

circle O at D, secant AOC intersects circle O at E, AE = 4,

AB = 12, and DB = 6. What is the length of OC? EXTERNAL times WHOLE secant = TANGENT times TANGENT

A

PO

BC

ew=tt

Tangent Intersecting a Secant

7) In the diagram below, tangent PA and secant PBC are

drawn to circle O from external point P. If PB = 4 and

BC = 5, what is the length of PA?

8) In the diagram below, tangent AB and secant ACD are

drawn to circle O from an external point A, AB = 8 and

AC = 4. What is the length of CD?

9) Secant ABC and tangent AD are drawn to circle O from

external point A. If AD = 10 and AC = 20, find the length

of AB.

DO NOW:Given: AB = 7, BC = 8, and AD = 3, find DE.

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Directions: Use the appropriate segment relationship to solve for

the missing measurement. If necessary, specify your answer in

simplest radical form.

1) Find x in the circle below.

2) Find x and the measure of ED in the circle below.

3) Given: AD = x, DE = 2, AB = 2, and CB = 10, find AD

and AE.

4) In circle O below, AE = 6, EB = 4, and CE = ED. Find CE

in simplest radical form.

A

B C

D

EO

5) In the circle below, find y.Then, find x to the nearest tenth.

6) In the diagram below of circle O, chords RT and QS intersect at M.

Secant PTR and tangent PS are drawn to circle O. The length of RM is

two more than the length of TM, QM = 2, SM = 12, and PT = 8. Find the

length of RT and PS.

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DO NOW:

A segment that touches a circle ONLY one time is

called a ____________________.

Fill in the blank below:

Theorems Involving Tangents To Circles

Common tangents: lines or segments that are tangent to more than one

circle at the same time

Tangents From The Same External Point-tangents to a circle from the same external point are equal.

(You may think of this as the "Hat"

Theorem because the diagram looks

like a circle wearing a pointed hat.)

Which two segments in

the circle above are equal?

Examples:1) What is the value of x in the circle below?

2) In the diagram below, circles X and Y have two tangents

drawn to them from external point T. The points of tangency are

C, A, S, and E. The ratio of TA to AC is 1:3. If TS = 24, find the

length of SE.

x28x

20

3) What is the value of x in the circle below?

What is the length of AB?

4) In the diagram below, triangle ABC is circumscribed about circle O

and the sides of triangle ABC are tangent to the circle at points D, E,

and F. If AB = 20, AE = 12, and CF = 12, what is the length of AC?

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5) In the diagram below, AB, BC, and AC are tangents to

circle O at points F, E, and D, respectively, AF = 6, CD = 5,

and BE = 4. What is the perimeter of triangle ABC?

Tangents Drawn To The Radius Of A Circle

If a line is tangent to a circle, it is perpendicular to the radius drawn

to the point of tangency.

If we connect points O and B, what type of triangle is formed?

Examples:

1) In the diagram below of triangle PAO, AP is tangent to

circle O at point A, OB = 7, and BP = 18. What is the

length of AP?

2) In the diagram below, AC and AD are tangent to circle B at

points C and D respectively, and BC, BD, and BA are drawn. If

AC = 12 and AB = 15, what is the length of BD?

DO NOW:

O

A

BC

D

In the diagram below, AB and CD are ________________.

OA and OB are ___________________.

Theorems About Chords and Circles-a radius perpendicular to a chord bisects the chord. (cuts chord in half)

-a radius that bisects a chord is perpendicular to the chord.

-the perpendicular bisector of a chord passes through the center of the circle.

If we connect points O and A we can

make a _____________________________.

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Examples:1) In the diagram below of circle O, diameter AOB is

perpendicular to chord CD at point E, OA = 6, and OE =2. What

is the length of CE, in simplest radical form?

2) In circle O shown below, diameter DB is perpendicular to chord

AC at E. If DB = 34, AC = 30, and DE > BE, what is the length of

BE?

3) In the diagram below of circle O, radius OC is 5cm. Chord AB is

8cm and is perpendicular to OC at point P. What is the length of

OP, in centimeters?

4) In circle O shown below, chords AB and CD and radius OA are

drawn, such that AB = CD, OE is perpendicular to AB, OF is

perpendicular to CD, OF = 16, CF = y + 10, and CD = 4y - 20.

Find the length of DF.

Determine the length of OA.

Another Theorem About Chords:

-the closer a chord is to the center of a circle, the

longer the chord.

5) In the diagram below, triangle ABC is inscribed in circle P. The

distances from the center of circle P to each side of the triangle are

shown.

Which statement about the sides of the triangle is true?

1) AB > AC > BC

2) AB < AC and AC > BC

3) AC > AB > BC

4) AC = AB and AB > BC

DO NOW:When lines are a constant distance apart from each

other, as shown in the diagram below, they are called

_______________________.

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Parallel ChordsIn a circle, parallel chords intercept congruent(equal) arcs.

Examples:1) In the diagram of circle O below, chord CD is parallel

to diameter AOB and mAC = 30. What is mCD?

2) In the diagram below of circle O, diameter AB is

parallel to chord CD. If mCD = 70, what is mAC?3) In the diagram of circle O below, chord CD is parallel to

diameter AOB and mCD = 110. What is mDB?

4) In circle O shown in the diagram below, chords AB and

CD are parallel. If mAB = 104 and mCD = 168, what is

mBD?

5) In the diagram below of circle O, chord AB is parallel to

chord CD. Which statement must be true?

is the symbol for CONGRUENT

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6) In the diagram below of circle O, chord AB ll chord CD,

and chord CD ll chord EF. Which statement must be true?

ll is the symbol for PARALLEL

7) In the diagram below, two parallel lines intersect

circle O at points A, B, C, and D with mAB = x + 20

and mDC = 2x - 20. Find x and mAB.

DONOW:

1.Nameacentralangle.

2.Nameaninscribedangle.

*USEYOURVOCABSHEET!*

CentralAngle:Acentralangleisanangleformedbytwointersectingradiisuchthatitsvertexisatthecenterofthecircle.

InscribedAngle:Aninscribedangleisananglewithitsvertex"on"thecircle,formedbytwointersectingchords.

Specialsituationsinvolvinginscribedangles:

Anangleinscribedinasemicircleisarightangle

Theoppositeanglesinacyclicquadrilateralaresupplementary.

Aquadrilateralinscribedinacircleiscalledacyclicquadrilateral.

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Examples:1) What is the value of x in the circle below?

2) Use what you know about central and inscribed angles

to find x in the circle below.

3) If the m

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Examples:

1) In the diagram below of circle O, chords AD and BC

intersect at E, the measure of arcAC = 87 and the measure of

arcBD = 35. What is the degree measure of

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7) In the accompanying diagram, AB is a diameter of

circle O and chord CD intersects diameter AB at E. If the

measure of arcAD = 100 and measure of arcAC = 40, find

m

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4) In the accompanying diagram of circle O, diameter AOB is extended

through B to external point P, tangent PC is drawn to point C on the

circle and the measure of arc AC : measure of arc BC = 7:2. Find

m

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Oct 25-2:41 PM DO NOW: Find the center and radius of each circle. 1) (x+ 8) 2 + y 2 = 4 2) x 2 + (y - 1) 2 = 6

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