notes chapter 09: circles - mathorama.com · notes chapter 09: circles unit 1: tangents, arcs, and...

Notes Chapter 09: Circles Unit 1: Tangents, Arcs, and Chords Section 1: Basic Terms

Definitions: A circle is the set of points in a plane that is equidistant from a given point. The distance is called the radius. The given point is the center.

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A A chord is a segment whose endpoints lie on a circle.

A secant is a line that contains a chord.

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

A tangent is a line in the plane of a circle that intersects the circle in exactly one point, called the point of tangency.

C

D

F O

E

G

T


Definitions: A sphere with center O and radius r is the set of all points in space at a distance r from point O.

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O

C

D

B

A

T

radii:

diameter:

chord:

secant:

tangent:

tangent segment:

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Definitions: 1.   Congruent circles (or spheres) are circles (or

spheres) that have congruent radii.

2.   Concentric circles are circles that lie in the same plane and have the same center. Concentric spheres are spheres that have the same center.

3.  A polygon is inscribed in a circle and circle is circumscribed about the polygon when each vertex of the polygon lies on the circle.

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Inscribed polygons

Circumscribed circles

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Examples on your desk

M N

O R Q

P N Point O is the center of the circle. Name each figure.

S

Q

R

U

T P 8.  Name a tangent to sphere S.

9.  Name a secant and a chord of the sphere.

10. Name two radii of the sphere (not drawn)

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Notes

on your desk Theorem 9.1 If a line is a tangent to a circle, then the line is perpendicular to the radius drawn to the point of tangency.

Chapter 09: Circles Unit 1: Tangents, Arcs, and Chords Section 2: Tangents

O

T Z m

Corollary Tangent segments to a circle from a point are congruent.

A

B

P

Theorem 9.2 If a line in the plane of a circle is perpendicular to a radius at its outer endpoint, then the line is tangent to the circle.

Q

R

m

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Notes

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When each side of a polygon is tangent to a circle, the polygon is said to be circumscribed about the circle and the circle is inscribed in the polygon.

Circumscribed polygons Inscribed circles

A line that is tangent to each of two coplanar circles is called a common tangent.

A common internal tangent intersects the segment joining the centers.

A common external tangent does not intersects the segment joining the centers.

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Notes

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A circle can be tangent to a line, but it can also be tangent to another circle. Tangent circles are coplanar circles that are tangent to the same line at the same point.

!A and !B are externally tangent.

B A

m m

D C

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9.7 !C and !D are internally tangent.

Notes

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Examples Name a line that satisfies the given description. 1.  Tangent to !P but not to !O

2.  Common external tangent to !O and !P.

3.  Common internal tangent to !O and !P.

O

A

E C D

P

B

Q

M

N

S

R P In the diagram, !M and !N are tangent at P. Segment PR and segment SR are tangents to !N. !N has diameter 16, PQ=3, and RQ=12 Complete. 5. PM = 6. MQ=

7. PR= 8. SR=

9. NS= 10. NR=

Notes

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Practice Name a line that satisfies the given

description. 1.  If OR=6 and OT=8, then RT=________.

2.  If m"OTR=45, and OT=4, then RT=______.

3.  What do you think is true about common internal tangents segment RS and segment TU? Prove your conjecture.

R

O

O

R

T S

P

U

T

V

Notes

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Chapter 09: Circles Unit 1: Tangents, Arcs, and Chords Section 3: Arcs and Central Angles

Minor Arc

A central angle of a circle is an angle with its vertex at the center of the circle.

Y

Z

Y

Z O O O W

Y

X

Z O

Y

X

Z

Major Arc Semicircles

The measure of minor arc is defined to be the measure of its central angle. Y

Z O

Minor Arc

50° 50°

Y

Z O

Major Arc

50°

W Y Z O

Semicircle

X

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Notes

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Adjacent arcs of a circle are arcs that have exactly one point in common.

C

O 50°

B

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Postulate 16 Arc Addition Postulate The measure of the arc formed by two adjacent arcs is the sum of the measures of these two arcs.

A

95°

C

O

B

A

80°

Congruent arcs of arcs, in the same circle or in congruent circles, that have equal measures.

80° Y

Z O

80°

D Name congruent arcs.

Notes

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Examples Name: 1.  two minor arcs

2.  two major arcs

3.  two semicircles

4.  an acute central angle

5.  two congruent arcs

R

A S

C

O

Give the measure of each angle or arc. 6. two minor arcs

7. two major arcs

8. two semicircles

X Y

W

Z

T

S

30°

Theorem 9.3 In the same circle or in congruent circles, two minor arcs are congruent if and only if their central angles are congruent.

Notes

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Chapter 09: Circles Unit 1: Tangents, Arcs, and Chords Section 4: Arcs and Chords

Theorem 9.4 In the same circle or in congruent circles: (1) Congruent arcs have congruent chords. (2) Congruent chords have congruent arcs.

T

A

U

R

O

C

Y

O

D B

Theorem 9.4 A diameter that is perpendicular to a chord bisects the chord and its arc.

S

Z

Notes

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Theorem 9.6 In the same circle or in congruent circles: (1) Chords equally distant from the center (or

centers) are congruent. (2) Congruent chords are equally distant from the

center (or centers).

T

U

R

O

y

5

B

S

13

Examples 1. x=_____, y=_____ 2. x=______, y=______ 3. x=______,

y=______

x

B x y 6 60°

y

x 8

Notes

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1.5

17

B

O

1. RT=_____, OM=_____ 2. 3.

M

C

C

O 220°

D

80°

Practice In the diagrams that follow, point O is the center of the circle. Complete.

R T

S A

O

Notes

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Chapter 09: Circles Unit 2: Angles and Segments Section 5: Inscribed Angles

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1

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

2

Theorem 9-7 The measure of an inscribed angle is equal to half the measure of its intercepted arc.

Notes

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Corollary 1 If two inscribed angles intercept the same arc, then the angles are congruent.

Corollary 2 An angle inscribed in a semicircle is a right angle.

Corollary 3 If a quadrilateral is inscribed in a circle, then its opposite angles are supplementary.

1 2

Theorem 9-8 The measure of an angle formed by a chord and a tangent is equal to half the measure of the intercepted arc.

130°

T

A P

Notes

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Examples Find the values of x and y in circle

O. 1. 

2. 

3. 

4. 

Notes

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Chapter 09: Circles Unit 2: Angles and Segments Section 6: Other Angles

Theorem 9-9 The measure of an angle formed by two chords that intersect inside a circle is equal to half the sum of the measures of the intercepted arcs.

1

D

C B

A

m"1=

Theorem 9-10 The measure of an angle formed by two secant and a tangent drawn from a point outside a circle is equal to half the difference of the measures of the intercepted arcs.

m"1= m"2= m"3=

1 2

3

Notes

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Example 1.  Segment UT is tangent to the

circle. Complete the following.

R

U

S

T V

W 100°

100°

30°

2.  Find m"D.

3. 

Notes

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Practice line AB is tangent to circle O; segment AF is a diameter;

1.  m"1= 2.  m"2= 3.  m"3= 4.  m"4= 5.  m"5= 6.  m"6= 7.  m"7= 8.  m"8=

B

O

G

F

E

D

C

A

100°

Warm Up

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Chapter 09: Circles Unit 2: Angles and Segments Section 7: Circles and Lengths of Segments

Walk-Around Problem: find the value of y.

Notes

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Theorem 9-11 When two chords intersects inside a circle, the product of the segments of one chord equals the product of the segments of the other chord. Show that r•s=t•u.

A

B

C D

P

Theorem 9-12 When two secant segments are drawn to a circle from the external point, the product of one segment and its external segment equals the product of the other secant segment and its external segment. Show that r•s=t•u.

A B

C D

P

Notes

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Theorem 9-13 When a secant segment and a tangent segment are drawn to a circle from an external point, the product of the secant segment and its external segment is equal to the square of the tangent segment. Show that r•s=t•u.

A B

D=C

P

Notes

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Practice Find the values of x. 1. 2. 3.

notes chapter 09: circles - mathorama.com · notes chapter 09: circles unit 1: tangents, arcs, and...

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