Geometry: Circles Terms and Theorems

Sector

piece of the circle

Segment

region enclosed by a chord

Arc

section of circumference

Tangent

line touches circle at one point (perpendicular to radius)

Chord

line segment whos endpoints are on circumference of the circle

Radius

distance from centerpoint to any edge (1/2 diameter)

Circumference

total distance around circles edge

Diameter

straight line segment passing through centerpoint and has endpoints on the circles edge

Secant

line intersecting a circle at 2 distinct points

Central Angle

angle whose vertex is exactly at the center

Inscribed Angle

angle formed by 2 chords in the circle that share a common endpoint on the circles edge

Central Angle Theorem

The measure of the arc formed by the endpoints of a central angle is [equal to the angle], m∠=marc

Arc Length Formula

Arc Length=x or measure of angle —-/360⋅2πr

Radian Measure

One radian is the measure of the angle that creates an arc the same length as the radius, 1 rad=180/π,1∘=π/180

Congruent Chords - Congruent Arcs

Two chords are congruent if and only if, Congruent chords → congruent arcs, Congruent arcs → congruent chords

Perpendicular Radius to Chord

If a diameter or radius is ⟂ to a chord, then it bisects the chord into 2 equal pieces, arc is cut into 2 equal arcs

Inscribed Angle Theorem

The measure of the inscribed angle is equal to half the measure of its intercepted arc, inscribed angle= ½ arc

Inscribed Angle Intercepting a Diameter

If an inscribed angle intercepts a diameter, then it is a right angle

Overlapping Arcs Theorem

If two inscribed angles intercept the same arc, then the angles are congruent

Inscribed Quadrilateral Theorem

If a quadrilateral is inscribed in a circle, then its opposite angles are supplementary

Interior Angle (Chords/ Secants Inside Circle)

The measure of the angle formed is equal to half the sum of the intercepted arcs, m∠=1/2(arc1+arc2)

On- the- Circle Angle (Tangent + Chord)

If a secant and a tangent intersect at the point of tangency, angle=1/2 arc

Exterior Angle (Secants/ Tangents Outside Circle)

The measure of the angle formed is equal to half the difference of the intercepted arcs, m∠=1/2(big arc−small arc)

Tangent perpendicular to radius

A line is tangent to a circle if it is perpendicular to the radius at the point of tangency

Two Tangents from same point

If two segments from the same external point are tangent, then they are congruent

Intersecting Chords (Inside)

a⋅b=c⋅d

Two Secants (Outside)

a(a+b)=c(c+d)

Secant- Tangent (Outside)

a² = b(b+c), tangent^2=secant(external)

Standard Form

(x−h)² + (y−k)² = r²

Area

πr^2

Circumference

2πr or πd