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Circles
Circles and Basic Terminology
I. Circle - the set of all points in a plane that are a given distance from a given point
(called the center) in the plane. Circles are named by their center.
II. Radius - a segment with one endpoint the center of a circle and the other endpoint on
the circle.
III. Circumference - the length of the perimeter of a circle; the distance around a circle.
IV. Chord - a segment that joins two points on a circle.
V. Diameter - a chord that passes through the center of the circle. The diameter of a given
circle is twice the length of its radius.
VI. Secant - a line that contains a chord of a circle.
VII. Tangent - a line in the plane of a circle that intersects the circle in exactly one point
(called "the point of tangency").
VIII. Congruent circles - circles that have congruent radii.
IX. Concentric circles - two circles in the same plane with the same center but different
radii.
X. Inscribed polygon - a polygon whose vertices are on the circle
XI. Circumscribed polygon - a polygon whose sides are tangent to a circle.
XII. Sphere - the set of all points in space a given distance from a point called the center
of the sphere. The given distance is called the radius of the sphere.
Properties of Tangents
I. Definitions A. Common tangent - a line that is tangent to two coplanar circles.
1. common external tangent - does not intersect the segment joining the
centers of the two circles.
2. common internal tangent - does intersect the segment joining the centers
of the two circles.
B. Externally tangent circles - if each circle lies in the exterior of the other.
C. Internally tangent circles - if one circle lies in the interior of the other.
II. Theorem - If a line is tangent to a circle, then it is perpendicular to the radius at the
point of tangency.
III. Theorem - Converse of Theo. 10.1
IV. Theorem - If two segments from a common exterior point are tangent to a circle, then
they are congruent.
V. Corollary - the line through an external point and the center of a circle bisects the
angle formed by the two tangents from the external point.
VI. Examples A. Example #1:
B. Example #2:
Angles and Arcs
I. Definitions A. Central angle - an angle whose vertex contains the center of a circle.
B. Arc - consists of two points and a continuous part of the circle that lies
between them.
1. major arc - an arc in the exterior of the central angle; by convention,
denoted by three letters (e.g., ACB).
<2. minoir arc - an arc which includes the cebtral angle; by convention,
denoted by two letters (e.g., AC).
C. Sum of Central Angles - the sum of the measures of the central angles of a
circle with no interior points in common is 360 degrees.
D. Semicircle - the congruent arcs formed when the diameter of a circle divides
the circle into two congruent arcs.
II. Postulate - the measure of adjacent, non-overlapping arcs is the sum of the measures
of the two arcs. That is,
III. Arc Measure versus Arc Length Arc Measure
major arc: 360
degrees - the degree
measure of its central
angle
minor arc: the degree
measure of the
central angle
Arc Length
the length of the circumference
proportional to the measure of the
central angle when compared to the
entire circle.
IV. Theorem - In the same or in congruent circles, two arcs are congruent IFF their
central angles are congruent.
Arcs and Chords
I. Arc of a chord - when a minor arc and a chord have the same endpoints.
II. Theorem - In a circle or congruent circles, two minor arcs are congruent IFF their
corresponding chords are congruent.
III. Theorem - In a circle, if a diameter is perpendicular to a chord, then the diameter
bisects the chord and its major and minor arcs.
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IV. Theorem – In a circle, if a diameter bisects a chords and its arcs, then it is
perpendicular to the chord
V. Theorem - If a chord is a perpendicular bisector of another chord, then it is a diameter.
VI. Theorem - In a circle or congruent circles, two chords are congruent if they are
equidistant from the center.
VII. Theorem – In a circle or congruent circles, if two chords are equidistant form the
center they are congruent
VIII. Example:
Sec. 10.5 - Inscribed Angles
I. Definitions A. Inscribed angle - An angle with vertex on a circle and sids that contain chords
of the circle.
B. Tangent-chord angle - an angle with vertex on a circle; one side tangent to the
circle at the vertex and the other side containing a chord.
II. Theorem - the measure of an inscribed angle is equal to one-half the measure of its
intercepted arc.
A. Example:
III. Theorem - If two inscribed angles of a circle or congruent circles intercept the same
or congruent arcs, then the angles are congruent.
IV. Theorem - An angle inscribed in a semicircle is a right angle; if an inscribed angle is
a right angle, its intercepted arc is a semicircle (and has degree measure of 180.)
V. Theorem - If a quadrilateral is inscribed in a circle, then its opposite angles are
supplementary.
VI. Example
Angles of Chords, Secants and Tangents
I. Theorem - The measure of a tangent-chord angle is one-half the measure of its
intercepted arc.
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II. Theorem - The measure of an angle formed by two chords intersecting in the interior
of a circle is one-half the sum of the intercepted arcs.
A. Example:
III. Theorem - The measure of an angle formed by two secants, two tangents, or a secant
and a tangent (drawn from a point on the exterior of a circle) is equal to the difference of
the measures of the intercepted arcs.
IV. Examples A. Example #1:
B. Example #2:
Extension - Special Segments in a Circle
I. Theorem - 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 segments of the second
chord.
A. Example:
II. Theorem - If two secant segments are drawn to a circle from an exterior point, then the
product of the lengths of one secant segment and its external segment equals the product
of the lengths of the other secant segment and its external segment.
A. Example:
III. Theorem - If a tangent segment and a secant segment are drawn to a circle from an
exterior point, then the square of the length of the tangent segment equals the product of
the lengths of the secant segment and is external segment.
A. Example:
IV. Circle/Segment Proof -
Equations of Circles
I. In the coordinate plane, the equation of a circle with center (h, k) and radius "r" can be
obtained by using the distance formula. Consider the circle below with center at the
origin and a radius of 6 units:
Now consider a circle with center at (-2, 3) and radius of 6 units:
This leads to the following which gives an equation of a circle that has its
center at (h, k) and radius of "r" units.
II. Examples
A. Example #1:
B. Example #2: