Chapter 10 Chapter 10 Properties of CirclesProperties of Circles

Date: Date: 3/18/113/18/11

Aim: Aim: 10.1 Use Properties of Tangents10.1 Use Properties of Tangents

CircleCircle

A set of all points in a plane that are A set of all points in a plane that are equidistant from a given point (the center).equidistant from a given point (the center).

RadiusRadius

Segment whose endpoints are the center Segment whose endpoints are the center and any point on the circle. and any point on the circle.

ChordChord

A segment whose endpoints are on a A segment whose endpoints are on a circle.circle.

DiameterDiameter

A chord that contains the center of a circle.A chord that contains the center of a circle.

SecantSecant

A line that intersects a circle in 2 points.A line that intersects a circle in 2 points.

TangentTangent

A Line in a plane of a circle that intersects A Line in a plane of a circle that intersects the circle in exactly one point, the circle in exactly one point, the point of the point of tangencytangency..

EXAMPLE 1Tell whether the line, ray, or segment is best described as a radius, chord, diameter, secant, or tangent of C.

ACa.

is a radius because C is the center and A is a point on the circle.

ACa.

b. AB is a diameter because it is a chord that contains the center C.

Tell whether the line, ray, or segment is best described as a radius, chord, diameter, secant, or tangent of C.

b. AB

c. DE is a tangent ray because it is contained in a line that intersects the circle at only one point.

Tell whether the line, ray, or segment is best described as a radius, chord, diameter, secant, or tangent of C.

DEc.

d. AE is a secant because it is a line that intersects the circle in two points.

Tell whether the line, ray, or segment is best described as a radius, chord, diameter, secant, or tangent of C.

AEd.

b. Diameter of A

Radius of Bc.

Diameter of Bd.

Use the diagram to find the given lengths.

a. Radius of A

a. The radius of A is 3 units.

b. The diameter of A is 6 units.

c. The radius of B is 2 units.

d. The diameter of B is 4 units.

EXAMPLE 3Tell how many common tangents the circles have and draw them.

a.

b.

c.

a. 4 common tangents

3 common tangentsb.

c. 2 common tangents

Chapter 10 Chapter 10 Properties of CirclesProperties of Circles

Date: Date: 3/21/113/21/11

Aim: Aim: 10.1 Use Properties of Tangents10.1 Use Properties of Tangents

Theorem Theorem In a plane, a line is tangent to a circle if In a plane, a line is tangent to a circle if

and only if the line is perpendicular to a and only if the line is perpendicular to a radius of the circle at its endpoint on the radius of the circle at its endpoint on the circle. circle.

Use the Converse of the Pythagorean Theorem. Because 122 + 352 = 372, PST is a right triangle and ST PT . So, ST is perpendicular to a radius of P at its endpoint on P. By Theorem 10.1, ST is tangent to P.

In the diagram, PT is a radius of P. Is ST tangent to P ?

In the diagram, B is a point of tangency. Find the radius r of C.

You know from Theorem 10.1 that AB BC , so ABC is a right triangle. You can use the Pythagorean Theorem.

AC2 = BC2 + AB2

(r + 50)2 = r2 + 802

r2 + 100r + 2500 = r2 + 6400

100r = 3900

r = 39 ft .

Pythagorean Theorem

Substitute.

Multiply.

Subtract from each side.

Divide each side by 100.

TheoremTheorem

Tangent segments from a common Tangent segments from a common external point are congruent.external point are congruent.

RS is tangent to C at S and RT is tangent to C at T. Find the value of x.

RS = RT

28 = 3x + 4

8 = x

Substitute.

Solve for x.

Tangent segments from the same point are

Is DE tangent to C?

Yes

ST is tangent to Q.Find the value of r.

r = 7

Find the value(s) of x.

+3 = x

Chapter 10 Chapter 10 Properties of CirclesProperties of Circles

Date: Date: 3/22/113/22/11

Aim: Aim: 10.2 Find Arc Measures 10.2 Find Arc Measures

Do Now: Do Now: Take out homeworkTake out homework

Vocabulary Vocabulary Central AngleCentral Angle

An angle whose vertex is the An angle whose vertex is the center of the circle. center of the circle.

Minor ArcMinor Arc If angle ACB is less than 180If angle ACB is less than 180°°

Major ArcMajor Arc Points that do not lie on the Points that do not lie on the

minor arc. minor arc.

Semi CircleSemi Circle Endpoints are the diameterEndpoints are the diameter

Measures Measures Measure of a Minor ArcMeasure of a Minor Arc

The measure of it’s central angle. The measure of it’s central angle.

Measure of a Major ArcMeasure of a Major Arc Difference between 360 and the measure of the Difference between 360 and the measure of the

minor arc. minor arc.

RSa. RTSb. RSTc.

RS is a minor arc, so mRS = m RPS = 110o.a.

RTS is a major arc, so mRTS = 360o 110o = 250o.b. –

Find the measure of each arc of P, where RT is a diameter.

c. RT is a diameter, so RST is a semicircle, and mRST = 180o.

Find Arc Measures

Arc Addition Postulate Arc Addition Postulate

The measure of an arc formed by two The measure of an arc formed by two adjacent arcs is the sum of the measures of adjacent arcs is the sum of the measures of the two arcs. the two arcs.

EXAMPLE 2

A recent survey asked teenagers if they would rather meet a famous musician, athlete, actor, inventor, or other person. The results are shown in the circle graph. Find the indicated arc measures.

a. mAC

a. mAC mAB= + mBC

= 29o + 108o

= 137o

Find Arc Measures

b. mACD

b. mACD = mAC + mCD

= 137o + 83o

= 220o

Identify the given arc as a major arc, minor arc, or

semicircle, and find the measure of the arc.

Examples

1. TQ

. QRT2

. TQR3

. QS4

. TS5

. RST6

minor arc, 120°

major arc, 240°

semicircle, 180°

semicircle, 180°

minor arc, 80°

minor arc, 160°

Congruent CirclesCongruent Circles Two circles are congruent if they have the same Two circles are congruent if they have the same

radius.radius.

Two arcs are congruent if they have the same Two arcs are congruent if they have the same measure and they are arcs of the same circle (or measure and they are arcs of the same circle (or congruent circles). congruent circles).

Are the red arcs congruent? Are the red arcs congruent?

Yes No

Chapter 10 Chapter 10 Properties of CirclesProperties of Circles

Date: Date: 3/23/113/23/11

Aim: Aim: 10.3 Apply Properties of Chords10.3 Apply Properties of Chords

Do Now: Do Now: Quiz Time.Quiz Time.

TheoremTheorem In the same circle, two minor arcs are congruent if In the same circle, two minor arcs are congruent if

and only if their corresponding chords are and only if their corresponding chords are congruent. congruent.

CDarcDCarcAB AB iff

EXAMPLE 1

In the diagram, P Q, FG JK , and mJK = 80o. Find mFG

So, mFG = mJK = 80o.

GUIDED PRACTICETry On Your Own

Use the diagram of D.

1. If mAB = 110°, find mBC

mBC = 110°

2. If mAC = 150°, find mAB

mAB = 105°

TheoremsTheorems If one chord is a perpendicular If one chord is a perpendicular

bisector of another chord, then bisector of another chord, then the first chord is a diameter. the first chord is a diameter.

If one diameter of a circle is If one diameter of a circle is perpendicular to a chord, then perpendicular to a chord, then the diameter bisects the chord the diameter bisects the chord and its arc. and its arc.

EXAMPLE 3Use the diagram of E to find the length of AC .

Diameter BD is perpendicular to AC . So, by the Theorem, BD bisects AC , and CF = AF.

Therefore, AC = 2 AF = 2(7) = 14.

Try On Your Own

1. CD

Find the measure of the indicated arc in the diagram.

mCD = 72°

2. DE

3. CEmCE = mDE + mCD

mCE = 72° + 72° = 144°

mCD = mDE.

mDE = 72°

TheoremTheorem In the same circle, two chords are congruent if In the same circle, two chords are congruent if

and only if they are equidistant from the center.and only if they are equidistant from the center.

In the diagram of C, QR = ST = 16. Find CU.

CU = CV

2x = 5x – 9

x = 3

So, CU = 2x = 2(3) = 6.

Use Theorem.

Substitute.

Solve for x.

1. QRQR = 32

Try On Your Own

2. URUR = 16

In the diagram, suppose ST = 32, and CU = CV = 12. Find the given length.

3. The radius of C

The radius of C = 20

Chapter 10 Chapter 10 Properties of CirclesProperties of Circles

Date: Date: 3/24/113/24/11

Aim: Aim: 10.4 Use Inscribed Angles and Polygons10.4 Use Inscribed Angles and Polygons

Do Now:Do Now:

Measure of an Inscribed Angle TheoremMeasure of an Inscribed Angle Theorem The measure of an inscribed angle is one half the The measure of an inscribed angle is one half the

measure of its intercepted arc. measure of its intercepted arc.

BAmADBm

2

1

inscribedangle

interceptedarc

A

B

D ●C

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

Example Example

RS

T Q

mQTS = 2m QRS = 2 (90°) = 180°

Theorem Theorem

If two inscribed angles of a circle intercept the same If two inscribed angles of a circle intercept the same arc, then the angles are congruent. arc, then the angles are congruent.

ECFmEDFm

F

ED

C

Find the measure of the red arc or angle.

1. m G = mHF = (90o) = 45o12

12

Try On Your Own

2. mTV = 2m U = 2 38o = 76o.

3. ZYN ZXN

ZXN 72°

Inscribed PolygonsInscribed Polygons A polygon is inscribed if all of its vertices lie on A polygon is inscribed if all of its vertices lie on

a circle. a circle. Circle containing the vertices is a Circle containing the vertices is a Circumscribed Circumscribed

CircleCircle. .

Theorem Theorem A A right triangleright triangle is inscribed in a circle is inscribed in a circle if and if and

only ifonly if the hypotenuse is a diameter of the the hypotenuse is a diameter of the circle. circle.

●C

Theorem Theorem A A quadrilateralquadrilateral is inscribed in a circle is inscribed in a circle if and if and

only ifonly if its opposite angles are supplementary. its opposite angles are supplementary.

●C

y = 105°x = 100°

1.

Find the value of each variable.

y = 112 x = 98

2.

c = 62 x = 10

Try On Your Own

Chapter 10 Chapter 10 Properties of CirclesProperties of Circles

Date: Date: 3/25/113/25/11

Aim: Aim: 10.5: Apply Other Angle Relationships in 10.5: Apply Other Angle Relationships in Circles Circles

Theorem Theorem If a tangent and a chord intersect at a point on If a tangent and a chord intersect at a point on

a circle, then the measure of each angle formed a circle, then the measure of each angle formed is one half the measure of its intercepted arc. is one half the measure of its intercepted arc.

BAmBAEm

2

1

ACBmBADm

2

1

Line m is tangent to the circle. Find the measure of the red angle or arc.

= 12 (130o) = 65o = 2 (125o) = 250ob. m KJLa. m 1

Find the indicated measure.

= 12 (210o) = 105om 1

Try On Your Own

= 2 (98o) = 196om RST

= 2 (80o) = 160om XY

Angles Inside the Circle TheoremAngles Inside the Circle Theorem

If If two chords intersect inside a circletwo chords intersect inside a circle, then the , then the measure of each angle is one half the measure of each angle is one half the sumsum of the of the measures of the arcs intercepted by the angle and its measures of the arcs intercepted by the angle and its vertical angles. vertical angles.

xo = 12

(mBC + mDA)x°

Find the value of x.

The chords JL and KM intersect inside the circle.

Use Theorem 10.12.xo = 12

(mJM + mLK)

xo = 12

(130o + 156o) Substitute.

xo = 143 Simplify.

Angles Outside the Circle TheoremAngles Outside the Circle Theorem If If a a tangent and a secanttangent and a secant,, two tangentstwo tangents,, or or two two

secantssecants intersect outside a circleintersect outside a circle, then the measure of , then the measure of the angle formed is the angle formed is one halfone half the the differencedifference of the of the measures of the intercepted arcs. measures of the intercepted arcs.

BABDAPm

2

1

Find the value of x.

Use Theorem 10.13.

Substitute.

Simplify.

The tangent CD and the secant CB intersect outside the circle.

= 12

(178o – 76o)xo

= 51 x

m BCD (mAD – mBD)= 12

Find the value of the variable.

y = 61o

Try On Your Own

= 104o a

5.

xo 253.7o

6.

Chapter 10 Chapter 10 Properties of CirclesProperties of Circles

Date: Date: 3/25/113/25/11

Aim: Aim: 10.6: Find Segment Lengths in Circles 10.6: Find Segment Lengths in Circles

Do Now:Do Now:

Segments of the ChordsSegments of the Chords If two chords intersect in the interior of the If two chords intersect in the interior of the

circle, then the product of the lengths of the circle, then the product of the lengths of the segments of one chord is equal to the product segments of one chord is equal to the product of the lengths of the segments of the other of the lengths of the segments of the other chord. chord.

EXAMPLE 1Find ML and JK.

NK NJ = NL NM Use Theorem

x (x + 4) = (x + 1) (x + 2) Substitute.

x2 + 4x = x2 + 3x + 2 Simplify.

4x = 3x + 2 Subtract x2 from each side.

x = 2 Solve for x.

ML = ( x + 2 ) + ( x + 1)

= 2 + 2 + 2 + 1

= 7

JK = x + ( x + 4)

= 2 + 2 + 4

= 8

Segments of Secants TheoremSegments of Secants Theorem If two secant segments share the same If two secant segments share the same

endpoint outside a circle, then the product of endpoint outside a circle, then the product of the lengths of one secant segment and its the lengths of one secant segment and its external segment equals the product of the external segment equals the product of the lengths of the other secant segment and its lengths of the other secant segment and its external segment. external segment.

RQ RP = RS RT Use Theorem 10.15.

4 (5 + 4) = 3 (x + 3) Substitute.

36 = 3x + 9 Simplify.

9 = x Solve for x

The correct answer is D.

Find x.

Find the value(s) of x.

13 = x

Try On Your Own.

x = 83 = x

Segments of Secants and Tangents Segments of Secants and Tangents TheoremTheorem

If a secant segment and a tangent segment If a secant segment and a tangent segment share an endpoint outside a circle, then the share an endpoint outside a circle, then the product of the lengths of the secant segment product of the lengths of the secant segment and its external segment equals the square of and its external segment equals the square of the length of the tangent segment. the length of the tangent segment.

Use the figure at the right to find RS.

256 = x2 + 8x

0 = x2 + 8x – 256

RQ2 = RS RT

162 = x (x + 8)

x –8 + 82 – 4(1) (– 256)

2(1)=

x = – 4 + 4 17

Use Theorem.

Substitute.

Simplify.

Write in standard form.

Use quadratic formula.

Simplify.

= – 4 + 4 17So, x 12.49, and RS 12.49

Find the value of x.

1.

x = 2

Try On Your Own.

2.

x = 245

3.

x = 8

1.

Try On Your Own.

2.

3.

Then find the value of x.

x = – 7 + 274

x = 8

x = 16

Chapter 10 Chapter 10 Properties of CirclesProperties of Circles

Date: Date: 3/25/113/25/11

Aim: Aim: 10.7: Write and Graph Equations of Circles10.7: Write and Graph Equations of Circles

Do Now:Do Now:

Standard Equation of a CircleStandard Equation of a Circle

222 rkyhx

Why?Why?

Write the equation of the circle shown.

The radius is 3 and the center is at the origin.

x2 + y2 = r2

x2 + y2 = 32

x2 + y2 = 9

Equation of circle

Substitute.

Simplify.

The equation of the circle is x2 + y2 = 9

Write the standard equation of a circle with center (0, –9) and radius 4.2.

(x – h)2 + ( y – k)2 = r2

(x – 0)2 + ( y – (–9))2 = 4.22

x2 + ( y + 9)2 = 17.64

Standard equation of a circle

Substitute.

Simplify.

C ● (0, -9)

4.2

Write the standard equation of the circle with the given center and radius.

1. Center (0, 0), radius 2.5x2 + y2 = 6.25

Try On Your Own.

2. Center (–2, 5), radius 7

(x + 2)2 + ( y – 5)2 = 49

EXAMPLE 3The point (–5, 6) is on a circle with center (–1, 3). Write the standard equation of the circle.

r = [–5 – (–1)]2 + (6 – 3)2

= (–4)2 + 32

= 5

(h, k) = (–1, 3) and r = 5 into the equation of a circle.

(x – h)2 + (y – k)2 = r2

[x – (–1)]2 + (y – 3)2 = 52

(x +1)2 + (y – 3)2 = 25

(x +1)2 + (y – 3)2 = 25.

Steps:1. Find values of h, k, and r by using the distance

formula.

2. Substitute your values into the equation for a circle.

212

212 yyxxd

1. The point (3, 4) is on a circle whose center is (1, 4). Write the standard equation of the circle.

The standard equation of the circle is(x – 1)2 + (y – 4)2 = 4.

Try On Your Own.

2. The point (–1 , 2) is on a circle whose center is (2, 6). Write the standard equation of the circle.

The standard equation of the circle is(x – 2)2 + (y – 6)2 = 25.

Graph A Circle

The equation of a circle is (x – 4)2 + (y + 2)2 = 36.

GRAPH

Rewrite the equation to find the center and radius.

(x – 4)2 + (y +2)2 = 36

(x – 4)2 + [y – (–2)]2 = 62

The center is (4, –2) and the radius is 6.

1. The equation of a circle is (x – 4)2 + (y + 3)2 = 16. Graph the circle.

Try On Your Own.

6. The equation of a circle is (x + 8)2 + (y + 5)2 = 121. Graph the circle.