# Chapter 9 Circles Define a circle and a sphere.

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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.

Bring a Compass Tomorrow

9.1 Basic Terms Objectives
Define and apply the terms that describe a circle.

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

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

The Circle B Point on circle A

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

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

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

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

Point of Tangency B Point of tangency A

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

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

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

Concentric Circles (or Spheres)
share the same center. G O Q

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.

Name each segment M Q O N R L P

OM M Q O N R L P

MN M Q O N R L P

MN M Q O N R L P

MQ M Q O N R L P

ML M Q O N R L P

ML M Q O N R L P

Point M M Q O N R L P

9.2 Tangents Objectives Apply the theorems that relate tangents and radii

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

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

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

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.

White Board Practice

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

Common External Tangents
B R A

Common External Tangents
B

Common Internal Tangents
B R A X

Common Internal Tangents
X R A B

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.

Remote Time How many common external tangents can be drawn?

Remote Time How many common external tangents can be drawn?

Remote Time How many common external tangents can be drawn?

Remote Time How many common external tangents can be drawn?

Remote Time How many common external tangents can be drawn?

Remote Time How many common external tangents can be drawn?

Remote Time How many common internal tangents can be drawn?

Remote Time How many common internal tangents can be drawn?

Remote Time How many common internal tangents can be drawn?

Remote Time How many common internal tangents can be drawn?

Remote Time How many common internal tangents can be drawn?

Remote Time How many common internal tangents can be drawn?

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

Externally Tangent Circles
B R A

Internally Tangent Circles
B R A

Remote Time Are the circles Externally Tangent Internally Tangent None

Remote Time Are the circles Externally Tangent Internally Tangent None

Remote Time Are the circles Externally Tangent Internally Tangent None

Remote Time Are the circles Externally Tangent Internally Tangent None

Remote Time Are the circles Externally Tangent Internally Tangent None

Remote Time Are the circles Externally Tangent Internally Tangent None

9.3 Arcs and Central Angles
Objectives Define and apply the properties of arcs and central angles.

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

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

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

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

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

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

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

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

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

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

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

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

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

White Board Practice Name two semicircles C R O A S

White Board Practice ARC, ASC C R O A S

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

White Board Practice AOR C R O A S

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

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

White Board Practice ARC, ASC C R O A S

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

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

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

Find the radius of the Arctic Circle
B A 66.6º 6400 W E O

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

Lecture 4 (9-4) Objectives
Define the relationships between arcs and chords.

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

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

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

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

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.

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

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

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

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

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

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

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.

9.6 Other Angles Objectives
Solve problems and prove statements involving angles formed by chords, secants and tangents.

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

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

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

9.7 Circles and Lengths of Segments
Objectives Solve problems about the lengths of chords, secants and tangents.

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

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

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