bell work what is a circle?. bell work answer a circle is a set of all points in a plane that are...

Bell work Bell work

What is a circle?What is a circle?

Bell work AnswerBell work Answer A A circlecircle is a set of all points in a is a set of all points in a

plane that are equidistant from a plane that are equidistant from a given point, called the given point, called the center center of the of the circlecircle

Unit 3 : Circles: Unit 3 : Circles: 10.1 Line & Segment Relationships 10.1 Line & Segment Relationships

to Circles (Tangents to Circles)to Circles (Tangents to Circles)

Objectives: Students will:Objectives: Students will:

1. Identify Segments and lines related 1. Identify Segments and lines related to circles. to circles.

2. Use Properties of a tangent to a 2. Use Properties of a tangent to a circle circle

Lines and Segments related to Lines and Segments related to circlescircles

•

CENTER

DIAMETERALSO A CHORD

RADIUSSECANT

CHORD

TANGENT LINE•

Exterior Point

•

Interior Point

Point of tangency


Center of the circleCenter of the circle

•

CENTER

P

Circle P


DiameterDiameter – from one point on the – from one point on the circle passing through the centercircle passing through the center(2 times the radius)(2 times the radius)

DIAMETERALSO A CHORD

•

CENTER


RadiusRadius – Segment from the center of – Segment from the center of the circle to a point on the circlethe circle to a point on the circle(1/2 the diameter)(1/2 the diameter)

•

CENTER

RADIUS (I)= 1/2 the Diameter


Chord – a segment from one point on Chord – a segment from one point on the circle to another point on the circlethe circle to another point on the circle

•DIAMETERALSO A CHORD

CHORD


SecantSecant – a – a line line passing through two passing through two points on the circlepoints on the circle

•

SECANT


TangentTangent – is a line that intersects the – is a line that intersects the

circle at exactly one pointcircle at exactly one point

•TANGENT LINE

•

Point of Tangency

Label Circle PartsLabel Circle Parts

1.1. SemicirclesSemicircles2.2. CenterCenter3.3. DiameterDiameter4.4. RadiusRadius

9. Tangent9. Tangent10. Secant10. Secant11. Minor Arc11. Minor Arc12. Major Arc12. Major Arc

5. Exterior6. Interior7. Diameter8. Chord

(p. 597)Theorem 10.1 (p. 597)Theorem 10.1

If a line is tangent to a circle, then it is If a line is tangent to a circle, then it is

perpendicularperpendicular ( _( _||_ )_ ) to the radius to the radius

drawn to the point of tangency.drawn to the point of tangency.

If line k is tangent to circle Q at point P,Then line k is _|_ toSegment QP.

Q

k

P

Tangent line

Example 1Example 1

Find the distance from Q to R, given Find the distance from Q to R, given

that line that line mm is tangent to the circle Q at is tangent to the circle Q at

Point P, PR = 4 cm and radius is 3 cm. Point P, PR = 4 cm and radius is 3 cm.

m

4 cm

P3 cm

R••

••Q

Example 1 answerExample 1 answer

Use the Pythagorean TheoremUse the Pythagorean Theorem

a² + b² = c²a² + b² = c²

3² + 4² = c²3² + 4² = c²

9 + 16 = c²9 + 16 = c²

√√25 = 25 = √√c²c²

c = 5c = 5

Example 2Example 2

Given that the radius (r) = 9 in,Given that the radius (r) = 9 in,

PR = 12, and QR = 16 in. Is the line PR = 12, and QR = 16 in. Is the line mm

tangent to the circle? tangent to the circle?

m

12 in

P9 in

R••

••Q

16 in

Example 2 answerExample 2 answer

No, it is not tangent. No, it is not tangent. Use the Pythagorean TheoremUse the Pythagorean Theorema² + b² = c²a² + b² = c²9² + 12² = 16²9² + 12² = 16²81 + 144 = 25681 + 144 = 256 225 = 256225 = 256

Since they are not = then the triangle is Since they are not = then the triangle is not a right triangle and thus the radius is not a right triangle and thus the radius is not perpendicular to the line m, therefore the not perpendicular to the line m, therefore the line is not tangent to the circle.line is not tangent to the circle.

Intersections of CirclesIntersections of Circles

No Points of IntersectionNo Points of Intersection

•CONCENTRIC CIRCLES –Coplanar circles that share a common center point


One Point of IntersectionOne Point of Intersection

•

•

The Circles are tangent to each other at the point

Common Tangents

Internal Tangent

External Tangent


Two Points of IntersectionTwo Points of Intersection

•

•

(p. 598) Theorem 10.3(p. 598) Theorem 10.3

If two segments from the same If two segments from the same

exterior point are tangent to a circle, exterior point are tangent to a circle,

then they are congruent. then they are congruent.

••

• • PP

••

••

R

T

__ __If SR and ST are tangent to circle P, __ __SR ST

S

Example 3Example 3

Segment SR and Segment ST are Segment SR and Segment ST are

tangent to circle P at Points R and T. tangent to circle P at Points R and T.

Find the value of x.Find the value of x.

••

• • PP

••

••

R

T

S

2x + 4

3x – 9

Example 3 AnswerExample 3 Answer ____ __ __Since SR and ST are tangent to the Since SR and ST are tangent to the circle, then the segments are circle, then the segments are , so , so

2x + 4 = 3x – 9 -2x -2x

4 = x – 9 + 9 + 9

13 = x

Unit 3 : Circles: Unit 3 : Circles: 10.2 Arcs and Chords10.2 Arcs and Chords


1. Use properties of arcs and chords to 1. Use properties of arcs and chords to

solve problems related to circles.solve problems related to circles.

Bell work Bell work Find the value of radius, x, if the Find the value of radius, x, if the diameter of a circle is 25 ft.diameter of a circle is 25 ft.

25 ft

x

Arcs of Circles Arcs of Circles CENTRAL ANGLECENTRAL ANGLE – An angle with its – An angle with its vertex at the center of the circlevertex at the center of the circle

Central Angle

60º•

CENTER P

P

A

B

•

•

Central Angle

60º•

CENTER P

P

A

B

•

•

Arcs of Circles Arcs of Circles

MINOR ARC

AB

C•MAJOR ARC

ACB

Minor Arc ABMinor Arc AB and and Major Arc ACBMajor Arc ACB


Central Angle

60º•

CENTER P

P

A

B

•

•

Measure of theMINOR ARC = the measure of theCentral Angle

AB = 60ºC •

The measure of the MAJOR ARC = 360 – the measure ofthe MINOR ARC ACB = 360º - 60º = 300º

TheThe measure of themeasure of the Minor Arc AB Minor Arc AB = the measure of the Central = the measure of the Central AngleAngle The measure of the The measure of the Major Arc ACB Major Arc ACB = 360= 360º - the measure of the º - the measure of the Central AngleCentral Angle

300º

Label Circle PartsLabel Circle Parts

1.1. SemicirclesSemicircles2.2. CenterCenter3.3. DiameterDiameter4.4. RadiusRadius

9. Tangent9. Tangent10. Secant10. Secant11. Minor Arc11. Minor Arc12. Major Arc12. Major Arc

5. Exterior6. Interior7. Diameter8. Chord


SemicircleSemicircle – an arc whose endpoints – an arc whose endpoints

are also the endpoints of the diameter are also the endpoints of the diameter

of the circle; of the circle; Semicircle Semicircle = 180= 180ºº

•

180º

Semicircle

Arc Addition PostulateArc 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 adjacent arcs is the sum of the measures of the two arcsmeasures of the two arcs

AB + BC = ABC

170º + 8 0º = 2 5 0º

•

••A

C

170º 80º

ARC ABC = 250º•

B

Example 1Example 1

Find m XYZ and XZFind m XYZ and XZ

••

••••X

Y

P

••Z

75º

110°


In the same circle or in congruent circles In the same circle or in congruent circles two minor arcs are congruent iff their two minor arcs are congruent iff their corresponding chords are congruentcorresponding chords are congruent

Congruent Arcs and Chords Congruent Arcs and Chords TheoremTheorem

Example 1: Given that Chords DE is Example 1: Given that Chords DE is

congruent to Chord FG. Find the value congruent to Chord FG. Find the value

of x.of x. Arc DE = 100º

Arc FG = (3x +4)º

D E

F G

Congruent Arcs and Chords Congruent Arcs and Chords TheoremTheorem

Example 2: Given that Arc DE is Example 2: Given that Arc DE is

congruent to Arc FG. Find the value congruent to Arc FG. Find the value

of x.of x.

Chord DE = 25 in

Chord FG = (3x + 4) in

D E

F G


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

••

Diameter

Chord

PCongruent Arcs

Congruent Segments


If one chord is the perpendicular bisector of another If one chord is the perpendicular bisector of another chord then the first chord is the diameter chord then the first chord is the diameter

••

Chord 1: _|_ bisectorof Chord 2, Chord 1 =the diameter

Chord 2

P

Diameter

(p. 606) Theorem 10.7(p. 606) Theorem 10.7In the same circle or in congruent circles, two In the same circle or in congruent circles, two chords are congruent iff they are equidistant from chords are congruent iff they are equidistant from the center. the center. (Equidistant means same perpendicular (Equidistant means same perpendicular distance)distance)

Chord TS Chord QR __ __iff PU VU

••P

Q

R

S

T

U

V

Center P

Example Example

Find the value of Chord QR, if TS = 20 Find the value of Chord QR, if TS = 20

inches and PV = PU = 8 inchesinches and PV = PU = 8 inches

••P

Q

R

S

T

U

V

8 in

8 in

Center P

Unit 3 : Circles: Unit 3 : Circles: 10.3 Arcs and Chords10.3 Arcs and Chords


1. Use inscribed angles and properties 1. Use inscribed angles and properties of inscribed angles to solve of inscribed angles to solve problems related to circlesproblems related to circles

Bell work 1Bell work 1Find the measure of Arc ABC, if Arc AB = 3x, Find the measure of Arc ABC, if Arc AB = 3x, Arc BC = (x + 80Arc BC = (x + 80º), and º), and __ ____ __AB AB BCBC

AB = 3xº

A

B

C

BC = ( x + 80º )

Bell work 2Bell work 2You are standing at point X. Point X is 10 You are standing at point X. Point X is 10

feet from the center of the circular water feet from the center of the circular water

tank and 8 feet from point Y. Segment XY is tank and 8 feet from point Y. Segment XY is

tangent to the circle P at point Y. What is the tangent to the circle P at point Y. What is the

radius, r,radius, r, of the circular water tank? of the circular water tank?

••

•• X

r 10 ft

8 ftY

P

Words for CirclesWords for Circles

1.1. Inscribed AngleInscribed Angle

2.2. Intercepted ArcIntercepted Arc

3.3. Inscribed PolygonsInscribed Polygons

4.4. Circumscribed Circumscribed CirclesCircles

Are there any words/terms that you are unsure of?

Inscribed AnglesInscribed AnglesInscribed angleInscribed angle – is an angle whose – is an angle whose vertex is on the circle and whose sides vertex is on the circle and whose sides contain chords of the circle.contain chords of the circle.

INSCRIBED ANGLE

A

B

INTERCEPTED ARC,

AB

Vertex on the circle

Intercepted ArcIntercepted ArcIntercepted ArcIntercepted Arc – is the arc that lies in – is the arc that lies in the interior of the the interior of the inscribed angleinscribed angle and and has endpoints on the angle. has endpoints on the angle.

INSCRIBED ANGLE

A

B

INTERCEPTED ARC,

AB

Vertex on the circle

(p. 613) Theorem 10. 8 (p. 613) Theorem 10. 8 Measure of the Inscribed AngleMeasure of the Inscribed Angle

The measure of an inscribed angle is equal The measure of an inscribed angle is equal half of the measure of its intercept arc.half of the measure of its intercept arc.

Central Angle

•

CENTER P

P

A

B

•

•Inscribed angle

C

m ∕_ ABC

= ½ m AC

m ∕_ ABC

= ½ mAC = 30º

Example 1Example 1

Central Angle

60º•

A

B

•

•

Measure of theINTERCEPTED ARC = the measure of theCentral Angle

AC = 60ºC

•

The measure of the inscribed angle ABC = ½ the The measure of the inscribed angle ABC = ½ the

measure of the intercepted AC.measure of the intercepted AC.

30º

Example 2Example 2

T

R

•

•U

•

Find the measure of the intercepted TU, if the Find the measure of the intercepted TU, if the

inscribed inscribed angle Rangle R is a right angle. is a right angle.

Example 3Example 3

T

R

•

•

TU = 86º

U

•

Find the measure of the inscribed angles Q , R ,and Find the measure of the inscribed angles Q , R ,and

S, given that their common intercepted TU = 86ºS, given that their common intercepted TU = 86º

Q

S

(p .614) Theorem 10.9(p .614) Theorem 10.9

T•

•

IF ∕_ Q and ∕_ S both

intercepted TU, then

∕_ Q ∕_ S

U

•

If two inscribed angles of a circle intercepted If two inscribed angles of a circle intercepted

the same arc, then the angles are congruentthe same arc, then the angles are congruent

Q

S

Inscribed vs. CircumscribedInscribed vs. Circumscribed

Inscribed polygonInscribed polygon – is when all of its – is when all of its vertices lie on the circle and the vertices lie on the circle and the polygon is inside the circle. The Circle polygon is inside the circle. The Circle then is then is circumscribedcircumscribed about the about the polygonpolygon

Circumscribed circleCircumscribed circle – lies on the – lies on the outside of the outside of the inscribed polygoninscribed polygon intersecting intersecting all the vertices of the polygon.all the vertices of the polygon.

Inscribed vs. CircumscribedInscribed vs. Circumscribed

The Circle is circumscribed about the The Circle is circumscribed about the polygon.polygon.

Circumscribed Circle

Inscribed Polygon


If a right triangle is inscribed in a If a right triangle is inscribed in a

circle, then the hypotenuse is the circle, then the hypotenuse is the

diameter of the circle. diameter of the circle.

•

Hypotenuse= Diameter

(p. 615) Converse of (p. 615) Converse of Theorem 10.10Theorem 10.10

If one side of an inscribed triangle is a If one side of an inscribed triangle is a diameter of the circle, then the triangle is a diameter of the circle, then the triangle is a right triangle and the angle opposite the right triangle and the angle opposite the diameter is a right angle. diameter is a right angle.

•

Diameter= Hypotenuse

B

The triangle is inscribed in the circle and one of its sides is the diameter Angle B is a right angleand measures 90º

Example Example

Triangle ABC is inscribed in the circle Triangle ABC is inscribed in the circle

Segment AC = the diameter of the Segment AC = the diameter of the

circle. Angle B = 3x. Find the value of x.circle. Angle B = 3x. Find the value of x.

•

B

A C

3xº


A quadrilateral can be inscribed in a circle iff its A quadrilateral can be inscribed in a circle iff its

opposite angles are supplementary.opposite angles are supplementary.

••

••••X

Y

P

••Z

The Quadrilateral WXYZ is inscribed in the circle iff

/ X + / Z = 180º, and

/ W + / Y = 180ºW

••

ExampleExample

A quadrilateral WXYZ is inscribed in circle P, if A quadrilateral WXYZ is inscribed in circle P, if ∕ ∕__ X = 103º and X = 103º and ∕_ ∕_ Y = 115º , Find the measures of Y = 115º , Find the measures of ∕ ∕__ W = ? and W = ? and ∕_ ∕_ Z = ? Z = ?

••

••••X

Y

P

••Z


/ X + / Z = 180º, and

/ W + / Y = 180ºW

••

103º 115º

ExampleExample

From Theorem 10.11From Theorem 10.11 ∕ ∕__ W = 180º – 115º = 65º and W = 180º – 115º = 65º and ∕ ∕__ Z = 180º – 103º = 77º Z = 180º – 103º = 77º

••

••••X

Y

P

••Z

The Quadrilteral WXYZ is inscribed in the circle iff

/ X + / Z = 180º, and

/ W + / Y = 180ºW

••

103º 115º

Unit 3 : Circles: Unit 3 : Circles: 10.4 Other Angle Relationships in 10.4 Other Angle Relationships in

CirclesCircles

Objectives: Students will:Objectives: Students will:1.1. Use angles formed by tangents and Use angles formed by tangents and

chords to solve problems related to chords to solve problems related to circlescircles

2.2. Use angles formed by lines Use angles formed by lines intersecting on the interior or intersecting on the interior or exterior of a circle to solve exterior of a circle to solve problems related to circlesproblems related to circles

Bell work 1Bell work 1

T

R

•

•

TU = 92º

U

•

Find the measure of the inscribed angles , R, Find the measure of the inscribed angles , R,

given that their common intercepted TU = 92ºgiven that their common intercepted TU = 92º



••

••••X

Y

P

••Z


/ X + / Z = 180º, and

/ W + / Y = 180ºW

••

130º 106º

(p. 621) Theorem 10.12 (p. 621) Theorem 10.12

If a tangent and a chord intersect at a point on a circle, then If a tangent and a chord intersect at a point on a circle, then the measure of each angle formed is ½ the measure of the measure of each angle formed is ½ the measure of its intercepted arc its intercepted arc

•P

A

B

•

Angle 1

C

m ∕_ 1

= ½ m minor AC

•

m ∕_ 2

= ½ m Major ABC

Angle 2

•

m

(p. 621) Theorem 10.12 (p. 621) Theorem 10.12 Example 1Example 1

Find the measure of Angle 1 and Angle 2, if Find the measure of Angle 1 and Angle 2, if the measure of the minor Arc AC is 130the measure of the minor Arc AC is 130ºº

•P

A

B

•

Angle 1

C

m minor AC = 130ºº

•Angle 2

•

m

(p. 621) Theorem 10.12 (p. 621) Theorem 10.12 Example 2Example 2

Find the measure of Angle 1, if Angle 1 = 6xFind the measure of Angle 1, if Angle 1 = 6xºº, and , and the measure of the minor Arc AC is (10x + 16)the measure of the minor Arc AC is (10x + 16)ºº

•P

A

B

•

Angle 1= 6xº

C

m minor AC = (10x + 16)ºº

•

•

m

Intersections of lines with respect to Intersections of lines with respect to a circlea circle

There are three places two lines can There are three places two lines can

intersect with respect to a circle.intersect with respect to a circle.

•

On the circle

•

In the circle Outside the cirlce

•


If two chords intersect in the If two chords intersect in the interiorinterior of a circle , then the of a circle , then the measure of each angle is ½ the measure of each angle is ½ the sumsum of the measures of of the measures of the arcs intercepted by the angle and its vertical angle.the arcs intercepted by the angle and its vertical angle.

•P

D

C•

Angle 1

B

m ∕_ 1

= ½ (m AB + m CD)

• m ∕_ 2

= ½ (mBC + mAD)

Angle 2

•

•A

m CD = 16º

(p. 622) Theorem 10.13(p. 622) Theorem 10.13Example Example


•P

D

C•

Angle 1

Bm AB = 40º

•

•

•A

xº

32

1

(p. 622) Theorem 10.14 (p. 622) Theorem 10.14 If a tangent and a secant, two tangents, or two secants If a tangent and a secant, two tangents, or two secants intersect in the intersect in the exteriorexterior of a circle, then the measure of the of a circle, then the measure of the angle formed is ½ the angle formed is ½ the difference difference of the intercepted arcs.of the intercepted arcs.

Bm ∕_ 1

= ½ (m BC – m AC)

•m ∕_ 2

= ½ (m PQR – m PR)

1 Tangentand 1 Secant

2 Tangents 2 Secants

•

•

••B

A

C

•

•

•

m ∕_ 3

= ½ (m XY – m WZ)

P

QR

••

••

••

W

X

Y

Z

xº

(p. 622) Theorem 10.14(p. 622) Theorem 10.14Example 1Example 1

Find the value of xFind the value of x

m ∕_ x

= ½ (m PQR - mPR)

Major Arc PQR= 266º

•

•

•

P

Q

R

•

(p. 622) Theorem 10.14(p. 622) Theorem 10.14Example 2 Example 2

Find the value of x, GFFind the value of x, GF.. The The mm EDG = 210 EDG = 210ºº

The The m m angle EHG = 68angle EHG = 68ºº

m ∕_ EHG = 68º

= ½ (m EDG – m GF)

•

•

•

•

E

H

G

xº

F

68ºD

Major Arc EDG= 210º

•

Unit 3 : Circles: Unit 3 : Circles: 10.5 Segment Lengths in Circles10.5 Segment Lengths in Circles


1.1. Find lengths of segments of chords, Find lengths of segments of chords, secants and tangentssecants and tangents

m CD = 20º



•P

D

C•

Angle 1

Bm AB = 65º

•

•

•A

xº

(p. 622) Theorem 10.14(p. 622) Theorem 10.14Example 2 Example 2

Find the value of x, GFFind the value of x, GF.. The The mm EDG = 300 EDG = 300ºº

The The m m angle EHG = 54angle EHG = 54ºº

m ∕_ EHG = 54º

= ½ (m EDG – m GF)

•

•

•

•

E

H

G

xº

F

54ºD


•

(p. 622) Theorem 10.14(p. 622) Theorem 10.14Example 2 Answer Example 2 Answer

m ∕_ EHG = 68º = ½ (m EDG – m GF) 54º = ½ ( 200º - xº ) 108º = 200º - xºxº = 200º - 108ºxº = 92º

m ∕_ EHG = 54º

= ½ (m EDG – m GF)

•

•

•

•

E

H

G

xº

F

54ºD


•



••

••••X

Y

P

••Z


/ X + / Z = 180º, and

/ W + / Y = 180ºW

••

130º 106º

Bellwork 2 AnswerBellwork 2 Answer

From Theorem 10.11From Theorem 10.11 ∕ ∕__ W = 180º – 106º = 74º and W = 180º – 106º = 74º and ∕ ∕__ Z = 180º – 130º = 50º Z = 180º – 130º = 50º

••

••••X

Y

P

••Z

The Quadrilteral WXYZ is inscribed in the circle iff

/ X + / Z = 180º, and

/ W + / Y = 180ºW

••

130º 106º

Segments of Chords, Secants and Segments of Chords, Secants and TangentsTangents

Segments of Chords

Segments of Secants

Segments of Tangents

External Segment


If two chords intersect in the interior of a circle, then the product of If two chords intersect in the interior of a circle, then the product of the lengths of the segments of one chord is equal to the product of the lengths of the segments of one chord is equal to the product of the lengths of the segments of the other chordthe lengths of the segments of the other chord

D

A

B

•

C

•

EA • EB = EC • ED •E

(p. 629) Theorem 10.15 (p. 629) Theorem 10.15 Example Example

Chord AB intersects Chord CD in the interior Chord AB intersects Chord CD in the interior of the circle at Point E, Find the measure of segment EA.of the circle at Point E, Find the measure of segment EA.Given ED= 6cm, EC = 16 cm, and EB = 8 cm. Given ED= 6cm, EC = 16 cm, and EB = 8 cm.

D

A

B

•

C

•

EA • EB = EC • ED •E 6 cm

16 cm ?

8 cm

(p. 629) Theorem 10.15 (p. 629) Theorem 10.15 Example AnswerExample Answer

EA = 12 cmEA = 12 cm

D

A

B

•

C

•

EA • EB = EC • EDEA • 8 = 16 • 68EA = 96 8 8

EA = 12 cm

•E 6 cm

16 cm

8 cm

12 cm


If two secant segments share the same endpoint outside the If two secant segments share the same endpoint outside the circle, then the product of the length of one secant segment circle, then the product of the length of one secant segment and the length of the of its external segment equals the and the length of the of its external segment equals the product of the length of the other secant segment and the product of the length of the other secant segment and the length of its external segment. length of its external segment.

EA • EB = EC • ED

E

A

B

C

D

(p. 630) Theorem 10.16 (p. 630) Theorem 10.16 ExampleExample

EA = 9 ft, AB = 11 ft, and EC = 10 ft. EA = 9 ft, AB = 11 ft, and EC = 10 ft. Find CD, the value of x and ED.Find CD, the value of x and ED.


E

A

B

C

D

9 ft

10 ft

11 ft

x


CD = x = 8 ftCD = x = 8 ftED = 18 ftED = 18 ft


9 (9+11) = 10(10 + x)9 (20) = 100 + 10x180 = 100 + 10x-100 -10080 = 10x10 10x = 8 ft and ED = (10+8)ED = 18 ft

E

A

B

C

D

9 ft

10 ft

11 ft

x


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

(EA)² = EC • ED

E

A

C

D

(p. 630) Theorem 10.17 (p. 630) Theorem 10.17 ExampleExampleEA = 9 in, CD = 15 in EA = 9 in, CD = 15 in

Find EC, the value of x.Find EC, the value of x.

(EA)² = EC • ED

E

A

C

D

x 15 in

9 in


Find EC, the value of x = 5, and ED= 20Find EC, the value of x = 5, and ED= 20

(EA)² = EC • ED

10² = x • (x + 15)

100 = x² + 15x

0 = x² + 15x - 100

x = -15 ± √15²- 4(1)(-100) 2 x = -15 + √625 = -15 +25

2 2x = (10/2) = 5

E

A

C

D

x 15 in

10 in

10.6 Equations of Circles10.6 Equations of Circles

TSW write the standard equation of a TSW write the standard equation of a circle given the radius and centercircle given the radius and center

TSW determine the radius and center TSW determine the radius and center of a circle given the equation in of a circle given the equation in standard formstandard form

Standard form of a CircleStandard form of a Circle

(x - (x - hh))22 + (y – + (y – kk))22 = = rr22

Center of Circle: (Center of Circle: ( h, h, kk))

Radius: Radius: rr

Example 1Example 1

Find the radius and center of the Find the radius and center of the given circle:given circle:

(x - (x - 33))22 + (y + 2) + (y + 2)22 = = 3636

xx2 2 + (y- 5)+ (y- 5)2 2 = 9= 9

Example 2Example 2

Write the equation of a circle given a Write the equation of a circle given a radius of 5 and center (3, -1)radius of 5 and center (3, -1)

Write the equation of a circle with a radius Write the equation of a circle with a radius of and a center (-1, 0)of and a center (-1, 0)7

bell work what is a circle?. bell work answer a circle is a set of all points in a plane that are...

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