Chapter 10 Flashcards
Circle
Set of all points in a plane that are equidistant from a given point called the center.
Interior of a circle
Points inside the circle.
Exterior of a circle
Points outside a circle.
Chord
A segment whose endpoints are on the circle.
Diameter
A chord that passes through the center of a circle.
Radius
Segment whose endpoints consist of the center of the circle and a point on the radius (Measure of the distance)
Tangent
If a line intersects a circle at exactly one point, the line is a tangent of a circle. This point of intersection is called the point of tangency.
Secant
If a line intersects a circle at two points, then the line is a secant. (Secants are ALWAYS lines, whereas tangent lines can be segments, lines, etc.)
Common Tangent
Tangent line that is tangent to two circles.
Common External Tangent
A common tangent that does not intersect the segment that join the centers of the circles.
Common Internal Tangent
A common tangent that intersects the segment that joins the centers of the circles.
Concentric Circles
Circles that have the same center. (Good for counterexamples)
Congruent Circles
Circles with congruent radii or diameters.
Theorem 10.1
If a line is tangent to a circle, then it is perpendicular to the radius drawn to the point of tangency.
Thoerem 10.2
In a plane, if a line is perpendicular to the radius of a circle at its endpoint on the circle, then the line is tangent to the circle.
Theorem 10.3
If two segments from the same exterior point are tangent to the circle, then they are congruent.
Inscribed
A circle is inscribed in a polygon if each side of the polygon is tangent to the circle.
Circumscribed
A circle is circumscribed about a polygon if each vertex of the polygon lies on the circle.
Central Angle
An angle who’s vertex is the center of the circle.
Minor Arc
Consists of the endpoints of the central angle and all points on the circle that are in the interior of the central angle.
Measure of a Minor Arc
The measure of the central angle measures less than 180.
Semicircle
An arc whose endpoints are t endpoints of the diameter.
Major Arc
Consists of the endpoints of the central angle and all points on the circle that lie in the exterior of the central angle.
Measure of a Major Arc
Always equal to 360- the measure of the associated minor arc.
Adjacent Arcs
Two arcs of the same circle are adjacent if they intersect at exactly one point.
Arc Addition Postulate
The measure of an arc formed by two adjacent arcs is the sum of the measures of the two arcs.
Congruent Arcs
In the same circle or in congruent circles, two arcs are congruent if they have the same measure.
Theorem 10.4
In the same circle or in congruent circles, two arcs are congruent if and only if the central angles are congruent.
Theorem 10.5
In the same circle or congruent circles, two minor arcs are congruent if and only if their corresponding chords are congruent.
Theorem 10.6
If a diameter of a circle is perpendicular to a chord, then the diameter bisects the chord and its arc.
Theorem 10.7
If chord AB is a perpendicular bisector of another chord, then AB is a diameter.
Theorem 10.8
In the same circle or in congruent circles, two chords are congruent if and only if they are equidistant from the center.
Inscribed Angle
An angle whose sides are chords of a circle.
Intercepted Arc
The arc that lies in the interior of an inscribed angle.
Theorem 10.9
If an angle is inscribed in a circle, then its measure is half the measure of its intercepted arc.
Theorem 10.10
If two inscribed angles of a circle intercept the same arc, then those angles are congruent. (THESE TRIANGLES ARE ALWAYS SIMILAR, SOMETIMES CONGRUENT.)
Theorem 10.11
An angle that is inscribed in a circle is a right angle if and only if its corresponding arc is a semicircle.
Theorem 10.12
A quadrilateral can be inscribed in a circle if and only if its opposite angles are supplementary.
