Circles Flashcards
tangent
line that intersects circle at one point only, perpendicular to radius from that point of intersection
secant
line that intersects circle at two points
2 types of arcs
minor: all points within segment/angle
major: all points in exterior of segment/angle
sector
formed by two radii and segment made by arc
segment
area inside segment made by arc and curved part of arc
major segment
everything outside of a segment
Arc Addition Postulate
If two arcs share an endpoint and no other points in common, then the two arcs and their measurements add up to equal the larger arc and its measurement.
Central Angle/Arc Postulate
If two central angles are equal, then their corresponding arcs are equal, and vice versa.
Arc/Chord Theorem
If two arcs are equal, their chords are equal, and vice versa.
Center/Perpendicular Theorem
A line drawn through the center of a circle perpendicular to a chord meets at its midpoint and bisects the arc determined by the chord.
2 Chord Theorems
In two congruent circles or in the same circle…
- If two chords are equal, the chords are the same distance from the center.
- if two chords are equally distant from the center, they are congruent.
External Point/Tangent Theorem
The tangents drawn to a circle from an external point are equal, and they form equal angles with the line joining the point to the center of the circle.
Tangent Circle Theorem
If two circles are tangent, their point of contact is on the line connecting their centers.
Center/Midpoint Theorem
The line joining the center of a circle to the midpoint of a chord is perpendicular to the chord.
Tangent and Radius Relationship
A tangent to a circle is perpendicular to the radius drawn to the point of tangency.
Line Joining Circle Centers Theorem
If two circles intersect, the line joining their centers is the perpendicular bisector of the common chord.
[photo] classification and measurement of ∠BAC
central angle; equal to intercepted arc (∠BAC = arc BC)
[photo] classification and measurement of ∠BDC
inscribed angle; half of intercepted arc (∠BDC = 1/2 arc BC)
[photo] classification and measurement of ∠BCD
chord and tangent angle; half of intercepted arc (∠BCD = 1/2 arc BC)
[photo] classification and measurement of ∠BEF (for example)
two chord angle; half the sum of intercepted arcs (∠BEF = 1/2 (arc BF + arc CD))
[photo] classification and measurement of ∠BDF
two secant angle; half the difference of intercepted arcs (∠BDF = 1/2 (arc BF - CE)
[photo] classification and measurement of ∠BDF
tangent and secant angle; half the difference of intercepted arcs (∠BDF = 1/2 (arc BE- arc CE))
[photo] classification and measurement of ∠BDF
two tangent angle; half the difference of intercepted arcs (∠BDF = 1/2 (arc CQE - arc CE))
Inscribed Angle Theorem Corollaries (4)
- Inscribed angles that intercept the same arc or equal arcs are equal.
- Angles inscribed in a semicircle are right angles.
- The opposite angles of a quadrilateral inscribed in a circle are supplementary.
- If a side of an inscribed quadrilateral is extended, the exterior angle formed is equal to the opposite interior angle.
Parallel Line Theorem
Parallel lines intercept congruent arcs on a circle.
Segment/Chord Theorem
If two chords of a circle meet at a point inside the circle, the product of the segments of one chord is equal to the product of the segments of the other.
Segment/Secant Theorem
If two secants of a circle meet at a point outside the circle, the product of one secant and its external segment is equal to the product of the other secant and its external segment.
Segment/Tangent/Secant Theorem
If a tangent and a secant are drawn to a circle from an external point, the square of the tangent is equal to the product of the secant and its external segment.
Cyclic Quadrilateral Theorem
If a quadrilateral is cyclic, then the opposite angles of the quadrilateral are supplementary. (and vice versa)
