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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. A B

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

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

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The Circle A B Point on circle

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

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

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

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

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Point of Tangency A B Point of tangency

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

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

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Congruent Circles (or Spheres) have equal radii. A D B E

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Concentric Circles (or Spheres) share the same center. O G 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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P M Q O N R L Name each segment

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P M Q O N R L OM

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P M Q O N R L MN

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P M Q O N R L

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P M Q O N R L MQ

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P M Q O N R L ML

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P M Q O N R L

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P M Q O N R L Point M

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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. A B tangent C Sketch

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Corollary Tangents to a circle from a common point are congruent. A X Y Z Sketch tangent

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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. A X B tangent

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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. A X B tangent R

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Common External Tangents A X B R

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A X B R

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Common Internal Tangents A X B R

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A X B R

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Construction 8 Given a point on a circle, construct the tangent to the circle through the point. Given: Construct: Steps:

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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. A B R

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Externally Tangent Circles A B R

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Internally Tangent Circles A B R

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Remote Time Are the circles A.Externally Tangent B.Internally Tangent C.None

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Remote Time Are the circles A.Externally Tangent B.Internally Tangent C.None

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Remote Time Are the circles A.Externally Tangent B.Internally Tangent C.None

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Remote Time Are the circles A.Externally Tangent B.Internally Tangent C.None

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Remote Time Are the circles A.Externally Tangent B.Internally Tangent C.None

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Remote Time Are the circles A.Externally Tangent B.Internally Tangent C.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. B A C

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

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

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Minor Arc an unbroken part of a circle with a measure less than 180°. B A C

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

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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. B A C D

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

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

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

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

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

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

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White Board Practice Name two semicircles R C S A O

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White Board Practice ARC, ASC R C S A O

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

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White Board Practice AOR R C S A O

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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. B A C D 90

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

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White Board Practice ARC, ASC R C S A O

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

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Group Practice m AB = 88 m BC = 52 m CD = 38 m DE = 104 m EA = 78 4x 3x 3x x 2x-14 A B C D E

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The radius of the Earth is about 6400 km O B A

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

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Find the radius of the Arctic Circle 6400 O B A EW 66.6º xº

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Find the radius of the Arctic Circle 6400 O B A EW 66.6º 23.4º

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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. B A D

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

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

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

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

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

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

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

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

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

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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. A D B C E 1

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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. A D B C E 1 F

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A O G F D E C B AB is tangent to circle O. AF is a diameter m AG = 100 m CE = 30 m EF = 25

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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. A D B X E F

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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. A D B C E 1 F 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. A D C E F Whole Piece Outside Piece = Whole Piece Outside Piece

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