CIRCLES--BASIC TERMS--CENTER, RADIUS
CIRCLE: the set of all points in a plane at a given distance from
a
given point in the plane.
CIRCLES--CHORD, SECANT, DIAMETER, TANGENT
CIRCLES--SPHERES
SPHERE: the set of all points
at a given distance from
a given point.
Many of the same
terms used with
circles are used with
spheres.
CIRCLES
CONGRUENT CIRCLES (or SPHERES)--
circles that have the same radii.
CONCENTRIC CIRCLES-- circles in the
same plane with the same center.
(also, concentric spheres)
POLYGON "INSCRIBED" IN A CIRCLE OR A CIRCLE "CIRCUMSCRIBED"
ABOUT A
POLYGON.
TANGENTS
THEOREM : If a line is tangent to a circle, then the line is perpendicular
to the radius drawn to the point of tangency.
COROLLARY: tangents to a circle from a
point are congruent.
CIRCLES--TANGENTS
POLYGON "INSCRIBED" IN A CIRCLE OR A CIRCLE "CIRCUMSCRIBED"
ABOUT A
POLYGON.
CIRCLE "INSCRIBED" IN A POLYGON OR POLYGON "CIRCUMSCRIBED"
ABOUT A
CIRCLE.
CIRCLES--TANGENTS
COMMON TANGENT--line tangent to two circles.
common " internal" tangent-- common "external" tangent.
intersects segment joining centers
TANGENT CIRCLES
"externally" "internally"
tangent. tangent.
CIRCLES--ARCS AND CENTRAL ANGLES
CIRCLES--ARC MEASURE.
CIRCLES--ARCS AND CENTRAL ANGLES
POSTULATE : ARC ADDITION POSTULATE:
CONGRUENT ARCS: arcs in the same circle, or in congruent circles, that have
the same measure.
CIRCLES--ARCS AND CENTRAL ANGLES
THEOREM : In the same or in congruent circles, two minor arcs are
congruent if and only if their central angles are congruent.
ARCS AND CHORDS
THEOREM : In the same circle or in congruent circles:
1. Congruent arcs have congruent chords.
2. Congruent chords have congruent arcs.
ARCS AND CHORDS
THEOREM : A diameter that is perpendicular to a chord bisects the
arc and
its chord.
ARCS AND CHORDS
THEOREM : 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).
INSCRIBED ANGLES
THEOREM : the measure of an inscribed angle equals one-half the
measure of its
intercepted arc.
INSCRIBED ANGLES
COROLLARY : If two inscribed angles intercept the same arc, then
the angles
are congruent.
COROLLARY : An angle inscribed in a semicirle is a ____________________
COROLLARY : The opposite angles of an inscribed quadrilateral are
supplementary.
INSCRIBED ANGLES
THEOREM : The measure of the angle formed by a chord and a tangent
equals one-half the measure of the intercepted arc.
OTHER ANGLES
THEOREM : The measure of the angle formed by two intersecting chords
equals half the sum of the intercepted arcs.
OTHER ANGLES
THEOREM : EXTERIOR CIRCLE INTERSECTIONS.
In each case, the measure of the angle formed equals half the
difference of the intercepted arcs.
CIRCLES AND LENGTHS OF SEGMENTS
THEOREM : When two chords intersect inside a circle:
CIRCLES AND LENGTHS OF SEGMENTS
THEOREM : When two secants intersect in the exterior of a circle:
CIRCLES AND LENGTHS OF SEGMENTS
THEOREM : When a tangent and a secant intersect in the exterior
of a
circle:
EQUATIONS OF CIRCLES
THEOREM : The equation of the circle with center ( a , b ) and
radius r is: