CHAPTER 9

Tangents, Arcs, and Chords

SECTION 9-1

Basic Terms

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

is a segment whose endpoints lie on a

circle.

CHORD

is a line that contains a chord.

SECANT

is a chord that contains the center of a circle.

DIAMETER

is a line in the plane of a

circle that intersects the circle in exactly one point, called the point

of tangency.

TANGENT

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

(center)

SPHERE

Are circles that have congruent radii

CONGRUENT CIRCLES

Are spheres that have congruent radii

CONGRUENT SPHERES

Are circles that lie in the same plane and have

the same center

CONCENTRIC CIRCLES

Are spheres that have the same center.

CONCENTRIC SPHERES

Occurs when each vertex

of a polygon lies on the circle and the circle is

CIRCUMSCRIBED about the POLYGON

INSCRIBED in a CIRCLE

SECTION 9-2

Tangents

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

Tangents to a circle from a point are congruent

Corollary

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

Occurs when each side of a polygon is tangent

to a circle and the circle is INSCRIBED in

the POLYGON

CIRCUMSCRIBED about the CIRCLE

a line that is tangent to each of two coplanar

circles

COMMON TANGENT

Intersects the segment joining the centers of

the circles.

COMMON Internal TANGENT

Does not intersect the segment joining the

centers of the circles.

COMMON External TANGENT

are coplanar circles that are tangent to the

same line at the same point

TANGENT CIRCLES

SECTION 9-3

Arcs and Central Angles

is an angle with its vertex at the center of

the circle.

CENTRAL ANGLE

is an unbroken part of a circle.

ARC

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

is the remaining arc formed by the

remaining points on the circle

* The measure is 360° minus the minor arc

MAJOR ARC

is an arc formed from the endpoints of a circle’s

diameter* The measure is 180°

SEMICIRCLE

Arcs that have exactly one point in common.

ADJACENT ARCS

Arcs in the same circle or in congruent circles

that have equal measure.

CONGRUENT ARCS

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

arcs

POSTULATE 16

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

SECTION 9-4

Arcs and Chords

In the same circle or in

congruent circles:1. Congruent arcs have

congruent chords2. Congruent chords have

congruent arcs

THEOREM 9-4

A diameter that is perpendicular to a

chord bisects the chord and its arc

THEOREM 9-5

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

SECTION 9-5

Inscribed Angles

Is an angle whose vertex is on a circle and whose sides contain chords of

the circle.

INSCRIBED ANGLE

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

INTERCEPTED ARC

The measure of an inscribed angle is equal to half the measure of

its intercepted arc

THEOREM 9-7

If two inscribed angles intercept the same arc,

then the angles are congruent

COROLLARY 1

An angle inscribed in a semicircle is a right

angle.

COROLLARY 2

If a quadrilateral is inscribed in a circle,

then its opposite angles are

supplementary

COROLLARY 3

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

SECTION 9-6

Other Angles

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

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

SECTION 9-7

Circles and Lengths of Segments

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

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

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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