UNIT 9: CIRCLES
Radius-Tangent Theorem
A line is a tangent to a circle is perpendicular to the radius of the circle intersecting the point of tangency
tangent segment
A segment of a tangent that has an endpoint at the point of tangency.
degree
A unit used to measure distances around a circle (360 degrees). One degree equals 1/360 of a full circle.
minor arc
An arc of a circle whose measure is less than 180 degrees.
circumference
C=2πr or C=πd, the distance around the circle (perimeter of a circle)
Congruent Arcs Theorem
Congruent Arcs have Congruent Central Angles: In the same circle, or in congruent circles, two minor arcs are congruent, then central angles of these arcs are congruent.
Congruent Central Angles Theorem
Congruent Central Angles have Congruent Chords: In the same circle, or on congruent circles, if 2 central angles are congruent, then their chords are congruent
Congruent Chords Theorem
Congruent Chords have Congruent Arcs: In the same circle, or in congruent circles, two chords are congruent, then the 2 arcs are congruent.
angle formed by a tangent and chord theorem
If a tangent to a circle and a chord intersect at a point on the circle, the the measure of each angle they form is one half the measure of its intercepted arc.
intercepted arc
The arc that lies in the interior of an inscribed angle and has endpoints on the angle.
arc addition postulate
The measure of an arc formed by two adjacent arcs is the sum of the measures of the two arcs
supplementary angles
Two angles whose sum is 180 degrees
complementary angles
Two angles whose sum is 90 degrees
diameter
a chord that passes through the center of the circle
center
a circle is named by its center, this circle is circle O because the circle center is point O.
similarity transformations
a composition of rigid transformations, or isometric, transformations with one or more dilations.
secant
a line or ray that intersects a circle at exactly two points
tangent to a circle
a line, line segment, or ray that intersects a circle at exactly one point and contains no points inside the circle
arc of a circle
a part of a circle between two given endpoint
radius
a segment that extends from the center of a circle to any point on the circle
chord
a segment that extends from the center of a circle to any point on the circle.
secant segment
a segment that intersects a circle twice and has one endpoint on the circle and one endpoint outside of the circle
radian
a standard unit of measure for angles; the measure of a central angle that subtends an arc that is equal in length to the radius of the circle
Second corollary to the inscribed angle theorem
an angle inscribed in a semicircle is a right angle.
inscribed angle
an angle whose vertex is on a circle and whose sides contain chords of the circle
circumscribed angle
an angle whose vertex is outside of a circle and whose sides are tangents to that circle
central angle
an angle whose vertex is the center of the circle and whose sides are radii of that circle
semi circle
an arc whose endpoints are the endpoints of a diameter
major arc
an arc whose measure is greater than or equal to 180 degrees
inscribed angle and central angle that have the same arc
an inscribed angle that intercepts the same arc as a central angle will be 1/2 of the measure of the central angle
concentric circles
circles that lie in the same plane and have the same center, so all circles are similar.
converse of the radius-tangent theorem
if a line, ray, or segment is perpendicular to a radius of a circle at the endpoint of the radius, then the line is tangent to the circle
The secant and tangent segment theorem
if a secant segment and a tangent segment share an endpoint outside a circle, then the product of the length of the secant segment and the length of its external segment equals the square of the length of the tangent segment.
Intersecting chords theorem
if two chords intersect inside a circle, then 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 chord.
angle formed by two chords theorem
if two chords intersect within a circle then the measure of each pair of vertical angles formed is equal to one half the sum of the measure of their intercepted arcs
secants and segments theorem
if two secant segments share the same endpoint outside a circle, then the product of the length of one secant segment and the length of its external segment equals the product of the length of the other secant segments and the length of its external segment.
arc length
s, is a portion of the circumference of a circle
central angle and its intercepted arc
the intercepted arc has the same measure as the central angle
angle formed by secants or tangents theorem
the measure of an angle formed by two secants, two tangents to a circle, or a secant and a tangent that intersect outside a circle is equal to half the difference of the measures of the arcs they intercept.
inscribed angle and its intercepted arc
the measure of the inscribed angle will be half the measure of the intercepted arc
Third corollary to the inscribed angle theorem
the opposite angles of a quadrilateral inscribed in a circle are supplementary
point of tangency
the point where a circle and a tangent intersect
sector of a circle
the region bounded by two radii of the circle and their intercepted arc
circle
the set of all points in a plane that are equidistant from a given point
First corollary to the inscribed angle theorem
two inscribed angles that intercept the same arc are congruent
Two Tangent Theorem
two segments tangent to a circle from a common external point are congruent
central angle and circumscribed angle
when a central angle and circumscribed angle intercepted the same arc or different congruent arcs, the central angle and circumscribed angle will be supplementary angles
inscribed angle and central angle with congruent arc
when a central angle and inscribed angle intercept the same arc or congruent arcs, the measure of the central angle will be twice the measure of the inscribed angle
