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Tangents, Arcs, and Chords
CHAPTER 9 Tangents, Arcs, and Chords
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SECTION 9-1 Basic Terms
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CIRCLE is the set of points in a plane at a given distance from a given point in that plane. The given point is the CENTER of the circle and the given distance is the RADIUS.
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is a segment whose endpoints lie on a circle.
CHORD is a segment whose endpoints lie on a circle.
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is a line that contains a chord.
SECANT is a line that contains a chord.
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is a chord that contains the center of a circle.
DIAMETER is a chord that contains the center of a circle.
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TANGENT is a line in the plane of a circle that intersects the circle in exactly one point, called the point of tangency.
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SPHERE is the set of all points in space at a distance r (radius) from point O (center)
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Are circles that have congruent radii
CONGRUENT CIRCLES Are circles that have congruent radii
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Are spheres that have congruent radii
CONGRUENT SPHERES Are spheres that have congruent radii
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Are circles that lie in the same plane and have the same center
CONCENTRIC CIRCLES Are circles that lie in the same plane and have the same center
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Are spheres that have the same center.
CONCENTRIC SPHERES Are spheres that have the same center.
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INSCRIBED in a CIRCLE Occurs when each vertex of a polygon lies on the circle and the circle is CIRCUMSCRIBED about the POLYGON
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SECTION 9-2 Tangents
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THEOREM 9 -1 If a line is tangent to a circle, then the line is perpendicular to the radius drawn to the point of tangency.
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Tangents to a circle from a point are congruent
Corollary Tangents to a circle from a point are congruent
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THEOREM 9 -2 If a line in the plane of a circle is perpendicular to the radius at its outer endpoint, then the line is tangent to the circle.
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CIRCUMSCRIBED about the CIRCLE
Occurs when each side of a polygon is tangent to a circle and the circle is INSCRIBED in the POLYGON
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a line that is tangent to each of two coplanar circles
COMMON TANGENT a line that is tangent to each of two coplanar circles
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COMMON Internal TANGENT
Intersects the segment joining the centers of the circles.
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COMMON External TANGENT
Does not intersect the segment joining the centers of the circles.
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TANGENT CIRCLES are coplanar circles that are tangent to the same line at the same point
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Arcs and Central Angles
SECTION 9-3 Arcs and Central Angles
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is an angle with its vertex at the center of the circle.
CENTRAL ANGLE is an angle with its vertex at the center of the circle.
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is an unbroken part of a circle.
ARC is an unbroken part of a circle.
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is the arc formed by two points on a circle
MINOR ARC is the arc formed by two points on a circle *The measure of a minor arc is the measure of its central angle
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is the remaining arc formed by the remaining points on the circle
MAJOR ARC is the remaining arc formed by the remaining points on the circle * The measure is 360° minus the minor arc
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is an arc formed from the endpoints of a circle’s diameter
SEMICIRCLE is an arc formed from the endpoints of a circle’s diameter * The measure is 180°
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Arcs that have exactly one point in common.
ADJACENT ARCS Arcs that have exactly one point in common.
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CONGRUENT ARCS Arcs in the same circle or in congruent circles that have equal measure.
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POSTULATE 16 The measure of the arc formed by two adjacent arcs is the sum of the measures of these two arcs
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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.
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SECTION 9-4 Arcs and Chords
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In the same circle or in congruent circles:
THEOREM 9-4 In the same circle or in congruent circles: Congruent arcs have congruent chords Congruent chords have congruent arcs
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THEOREM 9-5 A diameter that is perpendicular to a chord bisects the chord and its arc
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In the same circle or in congruent circles:
THEOREM 9-6 In the same circle or in congruent circles: Chords equally distant from the center (or centers) are congruent Congruent chords are equally distant from the center (or centers)
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SECTION 9-5 Inscribed Angles
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INSCRIBED ANGLE Is an angle whose vertex is on a circle and whose sides contain chords of the circle.
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Is the intersection of the sides of an inscribed angle and the circle
INTERCEPTED ARC Is the intersection of the sides of an inscribed angle and the circle
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THEOREM 9-7 The measure of an inscribed angle is equal to half the measure of its intercepted arc
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COROLLARY 1 If two inscribed angles intercept the same arc, then the angles are congruent
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An angle inscribed in a semicircle is a right angle.
COROLLARY 2 An angle inscribed in a semicircle is a right angle.
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COROLLARY 3 If a quadrilateral is inscribed in a circle, then its opposite angles are supplementary
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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.
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SECTION 9-6 Other Angles
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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.
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THEOREM 9-10 the measure of an angle formed by two secants, two tangents, or a secant and a tangent drawn from a point outside a circle is equal to half the difference of the measures of the intercepted arcs
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Circles and Lengths of Segments
SECTION 9-7 Circles and Lengths of Segments
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THEOREM 9-11 When two chords intersect inside a circle, the product of the segments of one chord equals the product of the segments of the other chord
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THEOREM 9-12 When two secant segments are drawn to a circle from an external point, the product of one secant segment and its external segment equals the product of the other secant segment and its external segment.
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THEOREM 9-13 When a secant 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
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