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Chapter 9 Circles. Define a circle and a sphere . Apply the theorems that relate tangent s, 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.
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Chapter 9Circles • 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.
9.1 Basic Terms Objectives • Define and apply the terms that describe a circle.
is a set of points in a plane equidistant from a given point. The Circle B A
The given distance is a radius (plural radii) The Circle B radius A
The given point is the center The Circle B radius A center
The Circle B Point on circle A
Chord any segment whose endpoints are on the circle. C B chord 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 B secant A
any line that contains exactly one point on the circle. Tangent B A tangent
Point of Tangency B Point of tangency A
is the set of all points equidistant from a given point. Sphere B A
Radii Diameter Chord Secant Tangent Sphere C E B A F D
have equal radii. Congruent Circles (or Spheres) B E D A
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.
L N M O Q R Name each segment P
L N M O Q R OM P
L N M O Q R MN P
L N M O Q R MN P
L N M O Q R MQ P
L N M O Q R ML P
L N M O Q R ML P
L N M O Q R Point M 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 tangent A 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.
Common Tangents are lines tangent to more than one coplanar circle. B tangent R A X
Common External Tangents X B R A
Common External Tangents R A X B
Common InternalTangents B R A X
Common InternalTangents X R A B
Construction 8 Given a point on a circle, construct the tangent to the circle through the point. Given: Construct: Steps:
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