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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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END

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