Chapter 10 - Geometry

circle

the set of all points in a plane that are equidistant from a given point, center of a circle.

tangent line to circle theorem

in a plane, a line is tangent to a circle if and only if the line is perpendicular to a radius of the circle at its endpoint on the circle

external tangent congruence theorem

tangent segments from a common external point are congruent

chord

a segment whose endpoints lie ON a circle

secant

line that intersects a circle at two points

tangent

line in the same place as a circle that intersects it at exactly 1 point

point of tangency

point where the tangent and a circle intersect

congruent cirlces

congruent radii

concentric circles

coplaner circles with common center

tangent circles

coplaner circles that intersect at 1 point only

common tangent

line tangent two two circles

central angle

an angle whose vertex is the center of the circle

arc

an unbroken piece of a circle consisting of two points (endpoints) and all points on the circle between them

minor arc

< 180

major arc

> 180

semicircle

= 180

arc addition postulate

the measure of an arc formed by two adjacent arcs is the sum of the measures of the two arcs.

congruent circles theorem

two circles are congruent circles if and only if they have the same radius

congruent central angles theorem

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

similar circles theorem

all circles are similar

congruent corresponding chords theorem

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

perpendicular chord bisector theorem

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

perpendicular chord bisector converse

if one chord of a circle is a perpendicular bisector of another chord, then the first chord is the diameter (defines diameter!)

equidistant chords theorem

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

always try to make…

RADIUS’ AND TRIANGLES!!!

inscribed angle

an angle whose vertex is on a circle and whose sides contain chords of the circle

inscribed arc

an arc that lies between two lines, rays, or segments

inscribed polygon

circle that contains a polygon is an inscribed polygon when all its vertices lie on a circle. the circle that contains the vertices is a circumscribed circle

measure of inscribed angle theorem

the measure of an inscribed angle is one-half the measure of its intercepted arc

inscribed angles of a circle theorem

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

inscribed right triangle theorem

if a right triangle is inscribed in a circle, then the hypotenuse is the diameter of the circle. Conversely, if one side of an inscribed triangle is a diameter of the circle, then the triangle is a right triangle and the angle opposite the diameter is the right angle

inscribed quadrilateral theorem

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

tangent and intersected chord theorem

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

angles inside the circle theorem

if two chords intersect INSIDE a circle, then the measure of each angle is 1/2 the SUM of the measure of the arcs intercepted by the angle and its vertical angle (opposite arcs)

angles outside the circle theorem

if a tangent and a secant, two tangents, or two secants intersect OUTSIDE a circle, then the measure of the angle formed is 1.2 the DIFFERENCE of the measures of the intercepted arcs (LOOK @ NOTES!!!!!!!!)

circumscribed angle

an angle whose sides are tangent to a circle

circumscribed angle theorem

the measure of a circumscribed angle is equal to 180 degrees minus the measure of the CENTRAL angle that intercepts the same arc

tangent segment

segment that is tangent to a circle at an endpoint

secant segment

segment that contains a chord and has one endpoint outside of the circle

external segment

part outside of the circle

segments of chords theorem

if two chords intersect in the interior of 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 (piece x piece = piece x piece)

segments of secants theorem

if two secant segments share the same endpoint outside a circle, then the product of the lengths of one secant segment and its external segment equals the product of the lengths of the other secant segment and its external segment (outside piece x whole = outside piece x whole)

segments of secants and tangents theorem

if a secant segment and a tangent segment share an endpoint outside a circle, then the product of the lengths of the secant segment and its external segment equals the square of the length of the tangent segment (tangent^2 = outside piece x whole)