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Circle Unit Summary Packet
10-1 Circles and Circumference
Vocabulary
pi ( π )- the ratio of circumference to diameter
for any circle
circle- a set of points in a plane that are
equidistant from one point
radius- a segment with endpoints at the center
of a circle and on the circle
inscribed- a polygon is inscribed in a circle if all
its vertices lie on the circle
chord- a segment with endpoints on the circle
circumscribed- a circle is circumscribed about a
polygon if the circle contains all the vertices of
the polygon
diameter- a chord that passes through the
center of a circle
concentric circles- circles with the same center
but different radii
circumference- the distance around a circle
10-2 Measuring Angles and Arcs
Vocabulary
central angle- an angle formed by two radii;
the vertex is the center of the circle
arc- an unbroken part of a circle
page 1
arc measure- the number of degrees in an arc
(same as the central angle that forms the arc)
adjacent arcs- arcs of the same circle that have
exactly one point in common
minor arc- an arc whose measure is less than
180 °
Arc Addition Postulate- the measure of an arc
formed by two adjacent arcs is the sum of the
measures of the two arcs
arc length- the distance along an arc measured
in linear units; it is a fraction of the
circumference
major arc- an arc whose measure is greater
than 180 °
semicircle- an arc whose measure is exactly
180 °
radian- the radian measure of a central
angle, θ, is the ratio of arc length to the radius
of the circle.
congruent arcs- arcs that have the same
measure
θ=
l
r
Theorems
10.1 (pg. 707)
In the same circle or in congruent circles, two minor arcs are congruent if and only if their central
angles are congruent.
page 2
10-3 Arcs and Chords
Vocabulary
distance from a point to a line- the segment
from the point that is perpendicular to the line
perpendicular bisector- a line that is
perpendicular to a segment and bisects the
segment
Theorems
10.2 (pg. 715)
In the same circle or in congruent circles, two minor arcs are congruent if and only if their
corresponding chords are congruent.
10.3 (pg. 716)
If a diameter (or radius) of a circle is perpendicular to a chord, then it bisects the chord and its arc.
10.4 (pg. 716)
The perpendicular bisector of a chord is a diameter (or radius) of the circle.
10.5 (pg. 717)
In the same circle or in congruent circles, two chords are congruent if and only if they are
equidistant from the center.
page 3
10-4 Inscribed Angles
Vocabulary
inscribed angle- an angle formed by two
chords; the vertex is on the circle
intercepted arc- an arc whose endpoints lie on
the sides of an inscribed angle
inscribed polygon- a polygon is inscribed in a
circle if all its vertices lie on the circle
Theorems
10.6 (pg. 723)
If an angle is inscribed in a circle, then the measure of the angle equals one half the measure of its
intercepted arc.
10.7 (pg. 724)
If two inscribed angles of a circle intercept the same arc or congruent arcs, then the angles are
congruent.
10.8 (pg. 725)
An inscribed angle of a circle intercepts a diameter or semicircle if and only if the angle is a right
angle.
page 4
10.9 (pg. 726)
If a quadrilateral is inscribed in a circle, then its opposite angles are supplementary.
10-5 Tangents
Vocabulary
tangent- a line that intersects a circle at
exactly one point
tangent segment- a segment of a tangent with
one endpoint on the circle
point of tangency- the point where a tangent
and a circle intersect
circumscribed polygon- a polygon is
circumscribed about a circle if each side of the
polygon is tangent to the circle
common tangent- a line that is tangent to two
circles
Theorems
10.10 (pg. 733)
In a plane, a line is tangent to a circle if and only if it is perpendicular to a radius drawn to the
point of tangency.
10.11 (pg. 734)
If two segments from the same exterior point are tangent to a circle, then the segments are
congruent.
page 5
10-6 Secants, Tangents, and Angle Measures
Vocabulary
secant- a line that intersects a circle at exactly two points
Theorems
10.12 (pg. 741)
If two secants intersect inside a circle, then the angle is half the sum of the two arcs (formed by
the angle and its vertical angle).
10.13 (pg. 742)
If a secant and a tangent intersect at the point of tangency, then the angle is half the arc.
10.14 (pg. 743)
If two secants, a secant and a tangent, or two tangents intersect outside the circle, then the angle
is half the difference of the two arcs.
page 6
10-7 Special Segments in a Circle
Vocabulary
chord segments- segments that form when two
chords intersect inside a circle
external secant segment- a secant segment
that lies in the exterior of a circle
secant segment- a segment of a secant line
that has exactly one endpoint on the circle
tangent segment- a segment of a tangent with
one endpoint on the circle
Theorems
10.15 (pg. 750)
If two chords intersect in a circle, then the products of the lengths of the chord segments of the
chords are equal.
10.16 (pg. 752)
If two secant segments intersect outside a circle, then the product of the measures of one secant
segment and its external secant segment is equal to the product of the measures of the other
secant segment and its external secant segment.
page 7
10.17 (pg. 752)
If a tangent segment and a secant segment intersect outside a circle, then the square of the
measure of the tangent segment is equal to the product of the measures of the secant segment
and its external secant segment.
10-8 Equations of Circles
Equation of a Circle
The equation of a circle with center (h, k) and radius r is (x – h)2 + (y – k)2 = r2
1. (x – 4)2 + (y – 3)2 = 52
3.
x2 + (y – 2)2 = 49
center is at ___________
center is at ___________
radius is ________ units
radius is ________ units
2. (x – 6)2 + (y + 5)2 = 9
4. (x + 3)2 + y2 = 16
center is at ___________
center is at ___________
radius is ________ units
radius is ________ units
page 8