The set of points in a plane at a given distance from a given point in that plane
Any segment that joins the center to a point of the circle is called a radius
A segment whose endpoints lie on a circle
A line that contains a chord
A chord that contains the center of a circle
A line in the plane of a circle that intersects the circle in exactly one point
Point of Tangency
The one point on the tangent that is touching the circle
Circles that have congruent radii
Circles that lie in the same plane and have the same center
Polygon inside the circle
Polygon outside the circle
A line is tangent to a circle
... then the line is perpendicular to the radius drawn to the point of tangency.
... to a circle from a point are congruent.
Perpendicular to a radius
If a line in the plane of a circle is .... at its outer endpoint, then the line is tangent to the circle.
A circle is an angle with its vertex at the center of the circle. Made up of two radii and equals the arc.
Two arcs; measure is always 180
Arcs that have exactly one point in common
Arc Addition Postulate
The measure of the arc formed by two adjacent arcs is the sum of the measures of these two arcs
... have congruent chords
Bisects the chord and the arc
A diameter that is perpendicular to a chord ...
Chords equally distant
...from the center are congruent.
An angle whose vertex is on a circle and whose sides contain chords of the circle
Measure of an inscribed angle
... is equal to half the measure of its intercepted arc
Two Inscribed Angles
... intercept the same arc, then the angles are congruent
An angle inscribed in a semicircle is a ....
If a ... is inscribed in a circle, opposite angles are supplementary.
The measure of an angle formed by a chord and a tangent is equal to HALF the measure of the INTERCEPTED arc.
The measure of an angle formed by two chords that intersect inside a circle is equal to HALF the SUM of the measures of the intercepted arcs
The measure of an angle formed by two secants, two tangents, or a secant and a tangent drawn from a point outside a circle is equal to HALF the DIFFERENCE of the measures of the intercepted arcs
When two chords intersect INSIDE a circle, the PRODUCT of the segments of one chord equals the PRODUCT of the segments of the other chord
When two secants are drawn to a circle from an EXTERIOR point, (sum of the whole secant)(exterior segment) = (sum of the other whole secant)(other exterior segment).
When a tangent is draw to a secant from an EXTERIOR point, (tangent)^2 = (Sum of the whole secant)(exterior segment)
tangent and chord intersecting on a circle
if a tangent and an chord intersect on a circle, then each angle formed is exactly 1/2 of its intercepted arc
if 2 chords intersect inside a circle, then each arc angle formed is 1/2 the sum of both the intercepted arcs.
secant/ tangent intersect outside the circle
if a secant & secant, tangent & tangent, or tangent and secant intersect outside the circle then the angle formed is 1/2 the difference of the intercepted arc.
A polygon that has all of its vertices on the circle
intersecting secant tangents
only use the quadratic formula when squaring the variable
if a line is a tangent to circle, then it is perpendicular to the radius drawn to the point of tangency
in a plane , if the a line is perpendicular to a radius of a circle at its endpoint on a circle, then the line is tangent to the circle
if to segments from the same exterior point are tangent to a circle, then they are congruent
in the same circle, or in congruent circles, two minor arcs are congruent if and only if their corresponding chords are congruent
if a diameter of a circle is perpendicular to a chord, then the diameter bisects the chord and its arc
if one chord is a perpendicular bisector of another chord, then the first chord is a diameter
in the same circle, or in congruent circles, two chords are congruent if and only if they are equidistant from the center
if an angle is inscribed in a circle, then its measure is half the measure of its intercepted arc
if two inscribed angles of a circle intercept the same arc, then the angles are congruent.
if a right triangle is inscribed in a circle, then the hypotenuse is a diameter of the circle. conversely if one side of an inscribed triangle is a diameter of a circle, then the triangle is a right triangle and the angle opposite the diameter is the right triangle.
a quadrilateral can be inscribed in a circle if and only if the opposite angles are supplementary
if a tangent and a chord intersect at a point on a circle then the measure of each angle formed is 1/2 the measure of its intercepted arc
if two chords intersect in the interior of the circle, then the measure of each angle is one half the sum of the measures of the arcs intercepted by the angle an its vertical angle
if a tangent and a secant, two tangents, are two secants intercept in the exterior of a circle, then the measure of the angle formed is 1/2 the difference of the measures of the intercepted arcs.
if two chords intersect in the interior of the circle then the product of the lengths of the segments of one chord is equal to the product of the length of the segments of the other chord
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 segment and the length of its external segment. (ea x eb= ec x ed)
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. (ea)squared= ec x ed