54 terms

Circle

The set of points in a plane at a given distance from a given point in that plane

Radius

Any segment that joins the center to a point of the circle is called a radius

Chord

A segment whose endpoints lie on a circle

Secant

A line that contains a chord

Diameter

A chord that contains the center of a circle

Tangent

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

Congruent Circles

Circles that have congruent radii

Concentric circles

Circles that lie in the same plane and have the same center

Inscribed

Polygon inside the circle

Circumscribed

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.

Tangents

... 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.

Central Angle

A circle is an angle with its vertex at the center of the circle. Made up of two radii and equals the arc.

Semicircles

Two arcs; measure is always 180

Adjacent arcs

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

Congruent 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.

Inscribed angle

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

Right Angle

An angle inscribed in a semicircle is a ....

Quadrilateral

If a ... is inscribed in a circle, opposite angles are supplementary.

Tang-Chord

The measure of an angle formed by a chord and a tangent is equal to HALF the measure of the INTERCEPTED arc.

Interior Angle

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

Exterior Angle

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

Chord Product

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

Secant Product

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).

Tangent Product

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

Intersecting chords

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.

inscribed polygon

A polygon that has all of its vertices on the circle

intersecting secant tangents

only use the quadratic formula when squaring the variable

Theorem 10.1

if a line is a tangent to circle, then it is perpendicular to the radius drawn to the point of tangency

Theorem 10.2

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

Theorem 10.3

if to segments from the same exterior point are tangent to a circle, then they are congruent

Theorem 10.4

in the same circle, or in congruent circles, two minor arcs are congruent if and only if their corresponding chords are congruent

Theorem 10.5

if a diameter of a circle is perpendicular to a chord, then the diameter bisects the chord and its arc

Theorem 10.6

if one chord is a perpendicular bisector of another chord, then the first chord is a diameter

Theorem 10.7

in the same circle, or in congruent circles, two chords are congruent if and only if they are equidistant from the center

Theorem 10.8

if an angle is inscribed in a circle, then its measure is half the measure of its intercepted arc

Theorem 10.9

if two inscribed angles of a circle intercept the same arc, then the angles are congruent.

Theorem 10.10

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.

Theorem 10.11

a quadrilateral can be inscribed in a circle if and only if the opposite angles are supplementary

Theorem 10.12

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

Theorem 10.13

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

Theorem 10.14

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.

Theorem 10.15

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

Theorem 10.16

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)

Theorem 10.17

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