This is a draft cheat sheet. It is a work in progress and is not finished yet.
Vocabulary
interior of a circle |
the set of all points inside the circle |
exterior of a circle |
the set of all points outside the circle |
chord |
a segment whose endpoints lie in a circle |
secant |
a line that intersects a circle at two points |
tangent |
a line in the same plane as a cicle that intersects it at exactly one point |
point of tangency |
the point where the tangent and a circle intersect is called the point of tangency |
common tangent |
a line that is tangent to two circles |
central angle |
an angle whose vertex is the center of a circle |
adjacent arcs |
arcs of the same circle that intersect at exactly one point |
congruent arcs |
two arcs within a circle or two circles that have the same measure |
sector of a circle |
a region bounded by two radii of the circle and their intercepted arc |
segment of a circle |
a region bounded by an arc and its chord |
arc length |
the distance along an arc measured in linear units |
inscribed angle |
an angle whose vertex is on a circle and whose sides contain chords of the circle |
intercepted arc |
consists of endpoints that lie on the sides of an inscribed angle and all the points of the circle between them |
subtends |
a chord or arc subtends an angle if its endpoints lie on the sides of the angle |
Formulas
*m = arc measurement in degrees* |
area of a sector of a circle |
A = 𝜋r2(m/360) |
area of a segment of a circle |
A = area of sector - area of the triangle formed inside the sector |
arc length |
L = 2𝜋r(m/360) |
Angle Relationships in Circles
vertex of the angle |
measure of angle |
on a circle |
half the measure of its intercepted arc |
inside a circle |
half the sum of the measures of its intercepted arcs |
outside a circle |
half the difference of the measures of its intercepted arcs |
Angle Relationships in Circles
vertex of the angle |
measure of angle |
on a circle |
half the measure of its intercepted arc |
inside a circle |
half the sum of the measures of its intercepted arcs |
outside a circle |
half the difference of the measures of its intercepted arcs |
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Pairs of Circles
congruent circles |
if and only if they have congruent radii |
concentric circles |
coplanar circles with the same center |
tangent circles |
two coplanar circle that intersect at exactly one point |
Arcs
minor arc |
an arc whose points are on or in the interior of a central angle |
the measure of a minor arc is equal to the measure of its central angle |
major arc |
an arc whose points are on or in the eterior of a central angle |
the measure of a major arc is equal to 360 degrees minus the measure of its central angle |
semicircle |
when the endpoints of an arc lie on a diameter |
the measure of a semicircle is equal to 180 degrees |
Theorems & Postulates
12-1-1 |
if a line is tangent to a circle, then it is perpendicular to the radius drawn to the point of tangency |
12-1-2 |
if a line is perpendicular to a radius of a circle, then the line is tangent to the circle |
12-1-3 |
if two segments are tangent to the same external point, then the segments are congruent |
12-2-3 |
in a circle, if a radius (or diameter) is perpendicular to a chord, then it bisects the chord and its arc |
12-2-4 |
in a circle, the perpendicular bisector of a chord is a radius (or diameter) |
12-4-1 inscribed angle theorem |
the measure of an inscribed angle is half the measure of its intercepted arc |
12-4-2 |
if inscribed angles of a circle intercept the same arc or are subtended by the same chord or arc then the angles are congruent |
12-4-3 |
an inscribed angle subtends a semicircle is and only if the angle is a right angle |
12-4-4 |
if a quadrilateral is inscribed in a circle, then its opposite angles are supplementary |
12-5-1 |
if a tangent and a secant (or chord) intersect on a circle at the point of tangency, then the measure of the angle formed is half the measure of its intercepted arc |
12-5-2 |
if two secants or chords intersect in the interior of a circle, then the measure of each angle formed is half the sum of the measures of its intercepted arcs |
12-5-3 |
if a tangent and a secant, two tangents, or two secants intersect in the exterior of a circle, then the measure of the angle formed is half the difference of the measures of its intercepted arcs |
Theorem 12-2-2
in a circle or congruent circles... |
1. congruent central angles have congruent chords |
2. congruent chords have congruent arcs |
3. congruent arcs have congruent central angles |
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