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**Chapter 9 Circles Define a circle and a sphere.**

Apply the theorems that relate tangents, chords and radii. Define and apply the properties of central angles and arcs.

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**Bring a Compass Tomorrow**

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**9.1 Basic Terms Objectives**

Define and apply the terms that describe a circle.

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The Circle is a set of points in a plane equidistant from a given point. B A

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The Circle The given distance is a radius (plural radii) B radius A

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The Circle The given point is the center B radius A center

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The Circle B Point on circle A

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Chord any segment whose endpoints are on the circle. C chord B A

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Diameter A chord that contains the center of the circle C B A diameter

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Secant any line that contains a chord of a circle. C secant B A

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**Tangent any line that contains exactly one point on the circle. B A**

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Point of Tangency B Point of tangency A

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Sphere is the set of all points equidistant from a given point. B A

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Sphere Radii Diameter Chord Secant Tangent C E B A F D

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**Congruent Circles (or Spheres)**

have equal radii. B E A D

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**Concentric Circles (or Spheres)**

share the same center. G O Q

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**Inscribed/Circumscribed**

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

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Name each segment M Q O N R L P

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OM M Q O N R L P

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MN M Q O N R L P

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MN M Q O N R L P

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MQ M Q O N R L P

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ML M Q O N R L P

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ML M Q O N R L P

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Point M M Q O N R L P

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9.2 Tangents Objectives Apply the theorems that relate tangents and radii

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Theorem If a line is tangent to a circle, then the line is perpendicular to the radius drawn to the point of tangency. B A tangent C Sketch

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**Corollary Tangents to a circle from a common point are congruent. Y**

X tangent Z Sketch

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

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**Inscribed/Circumscribed**

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.

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White Board Practice

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**Common Tangents are lines tangent to more than one coplanar circle. B**

X

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**Common External Tangents**

B R A

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**Common External Tangents**

B

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**Common Internal Tangents**

B R A X

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**Common Internal Tangents**

X R A B

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Construction 8 Given a point on a circle, construct the tangent to the circle through the point. Given: Construct: Steps: Do the construction for them on the board. The steps are in the textbook, and can be written down later.

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Remote Time How many common external tangents can be drawn?

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Remote Time How many common external tangents can be drawn?

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Remote Time How many common external tangents can be drawn?

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Remote Time How many common external tangents can be drawn?

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Remote Time How many common external tangents can be drawn?

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Remote Time How many common external tangents can be drawn?

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Remote Time How many common internal tangents can be drawn?

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Remote Time How many common internal tangents can be drawn?

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Remote Time How many common internal tangents can be drawn?

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Remote Time How many common internal tangents can be drawn?

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Remote Time How many common internal tangents can be drawn?

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Remote Time How many common internal tangents can be drawn?

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Tangent Circles are circles that are tangent to each other. B R A

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**Externally Tangent Circles**

B R A

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**Internally Tangent Circles**

B R A

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Remote Time Are the circles Externally Tangent Internally Tangent None

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Remote Time Are the circles Externally Tangent Internally Tangent None

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Remote Time Are the circles Externally Tangent Internally Tangent None

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Remote Time Are the circles Externally Tangent Internally Tangent None

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Remote Time Are the circles Externally Tangent Internally Tangent None

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Remote Time Are the circles Externally Tangent Internally Tangent None

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**9.3 Arcs and Central Angles**

Objectives Define and apply the properties of arcs and central angles.

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Central Angle is formed by two radii, with the center of the circle as the vertex. C A B

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Arc an arc is part of a circle like a segment is part of a line. A C B

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Arc Measure the measure of an arc is given by the measure of its central angle. 80 A C 80 B

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**Minor Arc an unbroken part of a circle with a measure less than 180°.**

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Semicircle an unbroken part of a circle that shares endpoints with a diameter. A C B

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Major Arc an unbroken part of a circle with a measure greater than 180°. B A C D

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Adjacent Arcs arcs that have exactly one point in common. D A C B

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**Arc Addition Postulate**

The measure of the arc formed by two adjacent arcs is the sum of the arcs. D A C B Sketch

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Congruent Arcs arcs in the same circle or in congruent circles that have the same measure. 90 90 D A C B

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White Board Practice Name two minor arcs C R O A S

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White Board Practice AR, RC, RS, AS, SC C R O A S

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White Board Practice Name two major arcs C R O A S

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White Board Practice ARS, ACR, RCS, RSA, RSC, CRS, CSR C R O A S

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White Board Practice Name two semicircles C R O A S

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White Board Practice ARC, ASC C R O A S

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White Board Practice Name an acute central angle C R O A S

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White Board Practice AOR C R O A S

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Theorem In the same circle or in congruent circles, two minor arcs are congruent only if their central angles are congruent. D 90 90 A C B

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White Board Practice Name two congruent arcs C R O A S

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White Board Practice ARC, ASC C R O A S

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**Group Practice Give the measure of each arc. D C 2x-14 4x 2x B E 3x**

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**Group Practice m AB = 88 m BC = 52 m CD = 38 m DE = 104 m EA = 78 D C**

2x-14 4x 2x B E 3x 3x + 10 A

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**The radius of the Earth is about 6400 km.**

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**The latitude of the Arctic Circle is 66. 6º North**

The latitude of the Arctic Circle is 66.6º North. That means the m BE 66.6º. B A 66.6º 6400 6400 W E O

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**Find the radius of the Arctic Circle**

xº B A 66.6º 6400 W E O

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**Find the radius of the Arctic Circle**

23.4º B A 66.6º 6400 W E O

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**Lecture 4 (9-4) Objectives**

Define the relationships between arcs and chords.

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Chord of the Arc The minor arc between the endpoints of a chord is called the arc of the chord, and the chord between the endpoints of an arc is the chord of the arc. D A B

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Theorem 9-4 In the same circle or in congruent circles, congruent arc have congruent chords and congruent chords have congruent arcs. D C A B Sketch

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Theorem 9-5 A diameter that is perpendicular to a chord bisects the chord and its arc. Y D X C A B Sketch

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Theorem 9-6 In the same circle or in congruent circles, chords are equally distant from the center only if they are congruent. D E Y B X C A Sketch

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**9.5 Inscribed Angles Objectives**

Solve problems and prove statements about inscribed angles. Solve problems and prove statements about angles formed by chords, secants and tangents.

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Inscribed Angle An angle formed by two chords or secant lines whose vertex lies on the circle. A C B

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Theorem The measure of an inscribed angle is half the measure of the intercepted arc. A C B

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Corollary If two inscribed angles intercept the same arc, then they are congruent. A D C B Sketch

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Corollary An angle inscribed in a semicircle is a right angle. A B O C

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Corollary If a quadrilateral is inscribed in a circle, then its opposite angles are supplementary. A B O D C

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Theorem An angle formed by a chord and a tangent has a measure equal to half of the intercepted arc. A B O C D

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Construction 9 Given a point outside a circle, construct the tangent to the circle through the point. Given: Construct: Steps: Do the construction for them on the board. The steps are in the textbook, and can be written down later.

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**9.6 Other Angles Objectives**

Solve problems and prove statements involving angles formed by chords, secants and tangents.

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Theorem The angle formed by two intersecting chords is equal to half the sum of the intercepted arcs. B C 1 A E D

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Theorem The angle formed by secants or tangents with the vertex outside the circle has a measure equal to half the difference of the intercepted arcs. B E C 1 A F D

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**AB is tangent to circle O**

AB is tangent to circle O. AF is a diameter m AG = 100 m CE = 30 m EF = 25 B D C 6 E 8 3 O 5 A F 7 2 1 4 G

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**9.7 Circles and Lengths of Segments**

Objectives Solve problems about the lengths of chords, secants and tangents.

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Theorem When two chords intersect, the product of their segments is equal. B E X A F D

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Theorem When two secant segments are drawn to a circle from a common point, the product of their length times their external segments is equal. B E C 1 A F D Whole Piece Outside Piece = Whole Piece Outside Piece

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Theorem When a secant and a tangent are drawn from a common point, the product of the secant and its external segment is equal to the tangent squared. E C A F D Whole Piece Outside Piece = Whole Piece Outside Piece

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