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CHAPTER 9 Tangents, Arcs, and Chords

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SECTION 9-1 Basic Terms

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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. CIRCLE

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is a segment whose endpoints lie on a circle. CHORD

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is a line that contains a chord. SECANT

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is a chord that contains the center of a circle. DIAMETER

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is a line in the plane of a circle that intersects the circle in exactly one point, called the point of tangency. TANGENT

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is the set of all points in space at a distance r (radius) from point O (center) SPHERE

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Are circles that have congruent radii CONGRUENT CIRCLES

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Are spheres that have congruent radii CONGRUENT SPHERES

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Are circles that lie in the same plane and have the same center CONCENTRIC CIRCLES

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Are spheres that have the same center. CONCENTRIC SPHERES

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Occurs when each vertex of a polygon lies on the circle and the circle is CIRCUMSCRIBED about the POLYGON INSCRIBED in a CIRCLE

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SECTION 9-2 Tangents

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If a line is tangent to a circle, then the line is perpendicular to the radius drawn to the point of tangency. THEOREM 9 -1

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Tangents to a circle from a point are congruent Corollary

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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. THEOREM 9 -2

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Occurs when each side of a polygon is tangent to a circle and the circle is INSCRIBED in the POLYGON CIRCUMSCRIBED about the CIRCLE

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a line that is tangent to each of two coplanar circles COMMON TANGENT

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Intersects the segment joining the centers of the circles. COMMON Internal TANGENT

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Does not intersect the segment joining the centers of the circles. COMMON External TANGENT

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are coplanar circles that are tangent to the same line at the same point TANGENT CIRCLES

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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

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is an unbroken part of a circle. ARC

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is the arc formed by two points on a circle * The measure of a minor arc is the measure of its central angle MINOR ARC

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is the remaining arc formed by the remaining points on the circle * The measure is 360° minus the minor arc MAJOR ARC

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is an arc formed from the endpoints of a circle’s diameter * The measure is 180° SEMICIRCLE

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Arcs that have exactly one point in common. ADJACENT ARCS

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Arcs in the same circle or in congruent circles that have equal measure. CONGRUENT ARCS

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The measure of the arc formed by two adjacent arcs is the sum of the measures of these two arcs POSTULATE 16

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In the same circle or in congruent circles, two minor arcs are congruent if and only if their central angles are congruent. THEOREM 9-3

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SECTION 9-4 Arcs and Chords

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In the same circle or in congruent circles: 1. Congruent arcs have congruent chords 2. Congruent chords have congruent arcs THEOREM 9-4

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A diameter that is perpendicular to a chord bisects the chord and its arc THEOREM 9-5

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In the same circle or in congruent circles: 1. Chords equally distant from the center (or centers) are congruent 2. Congruent chords are equally distant from the center (or centers) THEOREM 9-6

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SECTION 9-5 Inscribed Angles

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Is an angle whose vertex is on a circle and whose sides contain chords of the circle. INSCRIBED ANGLE

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Is the intersection of the sides of an inscribed angle and the circle INTERCEPTED ARC

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The measure of an inscribed angle is equal to half the measure of its intercepted arc THEOREM 9-7

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If two inscribed angles intercept the same arc, then the angles are congruent COROLLARY 1

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An angle inscribed in a semicircle is a right angle. COROLLARY 2

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If a quadrilateral is inscribed in a circle, then its opposite angles are supplementary COROLLARY 3

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The measure of an angle formed by a chord and a tangent is equal to half the measure of the intercepted arc. THEOREM 9-8

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SECTION 9-6 Other Angles

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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. THEOREM 9-9

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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 THEOREM 9-10

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SECTION 9-7 Circles and Lengths of Segments

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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 THEOREM 9-11

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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. THEOREM 9-12

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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 THEOREM 9-13

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END

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