53 terms

Concentric Circles

two or more coplanar circles with the same center

Congruent Circles

if two circles have congruent radii

Chord

a segment joining any two points on the circle

Diameter

a chord that passes through the center of the circle

Distance from the center to a chord =

the measure of the perpendicular segment from the center to the chord

(theorem) If a radius is perpendicular to a chord, then...

it bisects the chord

(theorem) if a radius of a circle bisects a chord that is not a diameter, then...

it is perpendicular to that chord

(theorem) The perpendicular bisector of a chord passes through...

the center of the circle

(theorem) if two chords of a circle are equidistant from the center, then...

they are congruent

(theorem) If two chords or a circle are congruent, then...

they are equidistant from the center of the circle

Arc

two points on a circle and all the points on a circle needed to connect the points by a single path

center of the arc

the center of the circle of which the arc is a part

Central Angle

an angle whose vertex is at he center of a circle

Minor Arc

an arc whose points are on or between the sides of a central angle (0 < x < 180)

Major Arc

an arc whose points are on or outside of a central angle (180 < x < 360)

Semicircle

an arc whose endpoints are the endpoints of a diameter (180)

Measure of a minor arc

the same a s the measure of the central angle that intercepts the arc

Measure of a major arc

360 minus the measure of the minor arc with the same endpoints

Congruent arcs

two arcs that have the same measure and are part of the same circle or congruent circle

(theorem) if two central angles of a circle (of of congruent circles) are congruent, then...

their intercepted arcs are congruent

(theorem) if two arcs of a circle (or of congruent circles) are congruent, then...

the corresponding central angles are congruent

(theorem) if two central angles of a circle (or of congruent circles) are congruent, then...

the corresponding chords are congruent

(theorem) if two chords or a circle (or of congruent circles) are congruent, then...

the corresponding central angles are congruent

(theorem) if two arcs of a circle (or of congruent circles), then...

they corresponding chords are congruent

(theorem) if two chords of a circle (or of congruent circles) are congruent, then...

the corresponding arcs are congruent

Secant

a line that intersects a circle at exactly 2 points (always contains a chord of the circle)

Tangent

a line that intersects a circle at exactly one point

Point of tangency (or point of contact)

the point where the line touches the circle

Facts about tangent lines in relation to radii

- a tangent line is perpendicular to the radius draw to the point of contact

- if a line is perpendicular to a radius at its outer endpoint, then it is tangent to the circle

- if a line is perpendicular to a radius at its outer endpoint, then it is tangent to the circle

Tangent Segment

the part of a tangent line between the point of contact and a point outside the circle

Secant Segment

the part of a secant line that joins a point outside the circle to the farther intersection point of the secant and the circle

External Part

the part of a secant line that joins the outside point to the nearer intersection point

(theorem) if two tangent segments are drawn from an external point, then...

those segments are congruent

Tangent Circles

circles that intersect each other at exactly one point

Externally Tangent Circles

if each of the tangent circles lies outside the other

Internally Tangent Circles

if one of the tangent circles lies inside the others

Line of Centers

connect the centers of two circles. For tangent circles, the point of contact lies on it.

Common tangent

a line tangent to two circles (not necessarily at the same point)

Common Internal Tangent

A common tangent that lies between the circles

Common External Tangent

A common tangent that is not between the circles

Central Angle

an angle whose vertex is the center of the circle (angle measure = arc measure)

Inscribed angle

Angle whose vertex is on the circle and sides are a tangent and chord that intersect at the tangent's point of contact

Tangent-chord angle

angle whose vertex is on the circle and sides are a tangent and a chord that intersect at the tangent's point of contact

(theorem) the measure of an inscribed angle or a tangent-chord angle (vertex on the circle) is...

one-half the measure of its intercepted arc

(angle measure = 1/2 arc measure)

(angle measure = 1/2 arc measure)

chord-chord angle

an angle formed by two chords that intersect inside a circle but not at the center

(theorem) the measure of a chord-chord angle is ______________________________ of the measures of the arcs intercepted by the chord-chord angle and its vertical angle.

1/2 the sum

Secant-secant angle

an angle whose vertex is outside a circle and whose sides are two secants

Secant-tangent angle

an angle whose vertex is outside a circle and sides are a secant and a tangent

Tangent-Tangent Angle

an angle whose vertex is outside a circle and sides are two tangents

(theorem) the measure of a secant-secant angle, a secant-tangent angle, or a tangent-tangent angle (vertex outside the circle) is...

one-half the difference of the measure of the intercepted arcs

(theorem) an angle inscribed in a semicircle...

is a right angle

length of arc

(marc/360) x 2(pi)r

Area of a sector of a circle

(n/360)(πr²), where n is the central angle.