Chapter 12: Circles Flashcards
Definition 60: A circle is
the set of all points in a plane that are at a given distance from a given point, called the center, in the plane.
Definition 61: A radius is
a line segment that joins the center of the circle to a point of the circle.
Postulate 17: All radii of the same circle are
congruent
Definition 62: A chord of circle is
a line segment that joins two points of the circle.
Definition 63: Diameter of a circle is
a chord that contains the center
Definition 64: A tangent line is
a line which interests a circle at exactly one point; the point of contact of called the point of tangency.
Definition 65: A secant line is
a line which interests a circle at two different points
Theorem 61: If a line through the center of a circle is perpendicular to a chord, it
also bisects the cord.
Theorem 62: In the same circle, congruent chords are equidistant (the same distance) from
the center of the circle.
Theorem 63: In the same circle, chords equidistant form the center of the circle are
congruent.
Theorem 64: If a radius is drawn to the point of tangency of a tangent line, then
the radius is perpendicular to the tangent line.
Theorem 65: If a radius is perpendicular to a line at the point where the line intersects a circle, then
the line is a tangent line.
Definition 66: A tangent segment is
a line segment that has a point on the tangent line and the point of tangency as an end point.
Theorem 66: If two tangent segments are drawn to a circle from the same exterior point, then
they are congruent.
Definition 67: An arc is
a portion of a circle consisting of two end points and the set of points on the circle that lie between those points.
Definition 68: A central angle is
an angle whose vertex is at the center of the circle.
Definition 69: The degree measure of a minor arc is the
measure of its central angle.
Definition 70: The circumference of a circle is the
distance around the circle, expressed in linear units of measurement (inches, centimeters, etc.).
Theorem 67: Arc Length-Degree Measure Proportion:
Arc Length/Circumference = Degree Measure/360
Postulate 18: Arc Addition Postulate: If P is on AB, then
mAP + mPB = mAB.
Definition 71: Congruent arcs are
arcs in the same or congruent circle which have the same degree measure.
Definition 72: The midpoint of an arc is the
point on the arc which divides the arc into two congruent arcs.
Theorem 68: If two chords of the same or congruent circles are congruent, then
their minor arcs are congruent.
Theorem 69: If two minor arcs of the same or congruent circles are congruent, then
their intersecting chords are congruent.
Theorem 70: In a circle, a diameter drawn perpendicular to a chord bisects the minor arc that
the chord intercepts.
Definition 73: An inscribed angle is
an angle whose vertex is on a circle and whose sides are chords (or secants) of the circle.
Theorem 71: An inscribed angle is
equal in measure to one-half the measure of its intercepted arc.
Corollary 71.1: Inscribed angles that intercept the same or congruent arcs are
congruent.
Corollary 71.2: An inscribed angle that intercepts a semicircle is a
right angle.
Theorem 72: The measure of an angle formed by a tangent and a chord drawn to the point of tangency is
equal to one-half the measure of the intercepted arc.
Theorem 73: The measure of an angle formed by two chords (or secants) intersecting in the interior of a circle is
equal to one-half the sum of the measures of the two intercepted arcs.
Theorem 74: The measure of an angle formed by two secants (or tangents) intersecting in the exterior of a circle is
equal to one-half the difference of the measures of the two intercepted arcs.
Theorem 75: If two chords intersect in the interior of a circle, the product of the lengths of the segments of one chord is
equal to the product of the lengths of the segments of the other.
Theorem 76: If two secant segments are drawn to a circle from the same exterior point, then the product of the lengths of one secant segment and its external segment is
equal to the product of the lengths of the other secant segment and its external segment.
Theorem 77: If a tangent segment and a secant segment are drawn to a circle from the same exterior point, then the square of the length of the tangent segment is
equal to the product of the lengths of the secant segment and its external segment.
