Download Circles and Circumference

Chapter 12 notes : Circles
Circles and Circumference
Circle: set of all points in a plane that are
equidistant from a given point (center).
Radius: segment from the center to a point on
the circle.
--congruent circles have congruent radii.
--all radii in a given circle are congruent.
Diameter: segment across a circle through the
center.
Chord: segment whose endpoints are on the
circle.
Circumference: the distance around a circle
pi: the ratio of Circumference to diameter
(22/7 or 3.14)
Angles and Arcs
Central Angle—an angle whose vertex is the
center of the circle.
Minor Arc—An arc on a circle with a central
angle less than 180 degrees.
Major Arc—An arc on a circle with a central
angle greater than 180 degrees.
Semicircle—An arc on a circle with a central
angle of exactly 180 degrees.
The measure of an arc is equal to the measure
of its central angle.
Theorem: In the same or congruent circles,
two arcs are congruent if and only if their
central angles are congruent.
(Arc Addition Postulate): The measure of an
arc formed by two adjacent arcs is the sum of
the measures of the two arcs
Arc Length: the length (using linear
measurement scale) of a particular arc by
treating it as part of the full circumference of
the circle
or
A = Arc Measure (in degrees)
= Arc length (in linear units...ft, cm, in...)
= Diameter of the circle
Arcs and Chords
Theorem: Two minor arcs in the same circle (or
congruent circles) are congruent if and only if
their corresponding chords are congruent.
Theorem: If a diameter (or radius) is
perpendicular to a chord, then it bisects the
chord and its arc.
Inscribed Polygons and Circumscribed Circles
Theorem: Two chords of a circle are congruent
if and only if they are equidistant from the
center.
Inscribed Angles
Inscribed Angle—an angle whose vertex is on a
circle and whose sides are chords.
Intercepted Arc—an arc on the interior of an
inscribed angle.
(Inscribed Angle Theorem): If an angle is
inscribed in a circle, then the measure of the
intercepted arc is twice the value of the
measure of the inscribed angle
- If two inscribed angles of a circle intercept
congruent arcs (or the same arc), then the
angles are congruent
If an inscribed angle intercepts a semicircle,
the angle is a right angle
-
If a quadrilateral is inscribed in a circle, then
its opposite angles are supplementary
-
Tangents
Tangent—a line that intersects a circle in
exactly one point.
Point of Tangency—the point where a tangent
line intersects a circle.
- If a line is tangent to a circle, then it is
perpendicular to the radius drawn to the point
of tangency
- If a line is perpendicular to a radius at its
endpoint on a circle, then the line is tangent to
the circle
- If two segments from the same exterior point
are tangent to a circle, then they are
congruent.
Secants, Tangents, and Angle
Measures
Secant—line that intersects a circle in two
points.
- If two secants (or chords) intersect in the
interior of a circle, then the measure of an
angle formed is 1/2 the sum of the measure of
the arcs intercepted by the angle and its
vertical angle (the angle is the average of the
intercepted arcs)
Angle = 1/2 * (Arc1 + Arc2)
- If a secant and a tangent intersect at a point
of tangency, then the measure of the angle is
1/2 the measure of the intercepted arc
Angle = 1/2 * intercepted arc
- If two secants, a secant and a tangent, or two
tangents intersect in the exterior of a circle,
then the measure of the angle is 1/2 * the
difference in the intercepted arcs.
Angle = 1/2 * (Arc1 - Arc 2)