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Larson Geometry Reference – Chapter 10
Definitions
Term/Concept
Adjacent Arcs
Arc - Degree Measure
Arc – Length
*Arc of a circle
Center
*Central Angle
Chord
Circle
*Circumference
Circumscribed Angle
Circumscribed Circle
Common Tangent
Concentric Circles
Congruent Arcs
Congruent Circles
*Diameter
External Segment
*Inscribed Angle
Inscribed Polygon
Intercepted Arc
Major Arc
*Minor Arc
*Point of Tangency
*Radius
*Secant Line
Secant Segment
Semicircle
Similar Arcs
Subtend
*Tangent Line
Tangent Segment
Description
Arcs of a circle that intersect at exactly one point.
The measure of a minor arc is the measure of its central angle.
The distance along the edge of the circle between the endpoints of the arc. Measured in linear units
such as feet, inches, cm, etc.
The points included in a portion of a circle.
The point that is equidistant from all points on the circle.
An angle whose vertex is at the center of a circle
A segment whose endpoints are on a circle
The set of all points in a plane that are equidistant from a point in the plane called the center .
The measure of the perimeter of a circle.
An angle whose sides are tangent to a circle.
A circle that contains the vertices of an inscribed polygon.
A line, ray or segment that is tangent to two or more circles.
Coplanar circles that have the same centers but different radii.
Two or more arcs are congruent if and only if they have the same measure and are arcs of the same or
congruent circles.
Two circles are congruent if and only if they have the same radius. Two circles are congruent if and
only if a rigid motion or composition of rigid motions maps one circle onto the other.
A chord that passes through the center of a circle.
The part of a secant segment that is outside the circle.
An angle whose vertex is on the circle and whose sides each contain chords of the circle.
A polygon in which all vertices lie on a circle.
An arc of a circle that is in the interior of a central angle or an inscribed angle.
An arc on a circle with a measure of greater than 180. The measure of a major arc is 360 minus the
measure of its central angle
An arc on a circle with a measure of less than 180. The measure of a minor arc is the measure of its
central angle
The point at which a tangent line intersects a circle.
The segment from the center of a circle to a point on a circle.
A line that intersects a circle in exactly two points.
The segment that contains a chord of a circle and has exactly one endpoint outside the circle.
An arc with endpoints that are endpoints of a diameter. An arc on a circle with a measure of 180.
The measure of a semicircle is a central angle of 180
Two arcs are similar if and only if they have the same measure. All congruent arcs are similar but not
all similar arcs are congruent. (Two circles can have different radii but arcs of equal measure that
would not be congruent)
If the endpoints of a chord or arc lie on the sides of an inscribed angle, then the chord or arc is said to
subtend the angle.
A line in the plane of the circle that intersects the circle at only one point.
A segment that is tangent to a circle at an endpoint.
Formulas
Term/Concept
*Equation of a Circle
Description
In general, an equation for a circle with center at (h, k) and a radius of r units is
( x  h) 2  ( y  k ) 2  r 2
Postulates
Arc Addition Postulate
The measure of an arc formed by two adjacent arcs is the sum of the measures of the two arcs. If Q is a
point on PR then mPR + mQR = mPQR
Larson Geometry Reference – Chapter 10
Theorems
Theorems
*Angles Inside a Circle
Theorem
*Angles Outside a Circle
Theorem
Circumscribed Angle
Theorem
Congruent Corresponding
Chords Theorem
Congruent Circles Theorem
Congruent Central Angles
Theorem
Equidistant Chords Theorem
External Tangent Congruence
Theorem
Inscribed Angles of a Circle
Theorem
Inscribed Right Triangle
Theorem
*Inscribed Quadrilateral
Theorem
*Measure of an Inscribed
Angle Theorem
Perpendicular Chord Bisector
Converse Theorem
*Perpendicular Chord
Bisector Theorem
*Segments of Chords
Theorem
*Segments of Secants
Theorem
*Segments of Secants and
Tangents Theorem
Similar Circles Theorem
Tangent and Intersected
Chord Theorem
*Tangent Theorem
If two chords intersect inside a circle, then the measure of each angle formed is one-half the sum of
the measures of the arcs intercepted by the angle and its vertical angle.
If two secants, a secant and a tangent, or two tangents intersect outside the a circle, then the measure
of the angle formed is one-half the positive difference of the measures of the intercepted arcs.
The measure of a circumscribed angle is equal to 180 minus the measure of the central angle that
intercepts the same arc.
In a circle or in congruent circles, two minor arcs are congruent if and only if their corresponding
chords are congruent.
Two circles are congruent if and only if they have the same radius.
In the same circle, or in congruent circles, two minor arcs are congruent if and only if their
corresponding central angles are congruent.
In the same circle or in congruent circles, two chords are congruent if and only if they are
equidistant from the center.
If two segments from the same exterior point are tangent to a circle then they are congruent.
If two inscribed angles of a circle intercept the same arc, then the angles are congruent.
If a right triangle is inscribed in a circle, then the hypotenuse is a diameter of the circle. Conversely,
if one side of an inscribed triangle is a diameter, then the triangle is a right triangle and the angle
opposite the diameter is the right angle.
A quadrilateral can be inscribed in a circle if and only if its opposite angles are supplementary. If a
quadrilateral is inscribed in a circle, then its opposite angles are supplementary.
The measure of an inscribed angle equals one-half the measure of its intercepted arc.
If one chord is a perpendicular bisector of another chord, then the first chord is a diameter.
In a circle, if a diameter is perpendicular to a chord, then it bisects the chord and its arc.
If two chords intersect in a circle, then the product of the lengths of the segments of one chord
equals the product of the lengths of the other chord.
If two secant segments have the same endpoint outside a circle, then the product of the lengths of
one secant segment and its external segment is equal to the product of the lengths of the other secant
segment and its external segment.
If a tangent segment and a secant segment are have the same endpoint outside a circle, then the
square of the length of the tangent segment is equal to the product of the lengths of the secant
segment and its external segment.
All circles are similar
If a tangent and a chord intersect at a point on a circle, then the measure of each angle formed is one
half the measure of its intercepted arc.
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. If a line is tangent to a circle, then it is perpendicular to the radius draw
to the point of tangency.