Transcript Circles_powerpoint_v2

Circle
Is the set of all points equidistant
from a given point called the center.
The man is the
center of the circle
created by the shark.
Parts of a circle
We name a circle using its center point.
⊙ W is shown
W•
So, the Center is defined as the point inside the circle equidistant from all
points on the circle.
Diameter: Any line segment (chord) that contains the
circle’s center. BE (ALL diameters of a given circle are  )
Radius:
Any line segment with 1 endpoint at the
center of the circle, and the other endpoint ON the
circle. XF (The Radius is = ½ diameter. All radii of a given circle are )
XF  XE  XB
B .
. X
.E
F
Chord: Any line segment with endpoints that lie on the circle.
CD
Secant: Any line that intersects the circle in exactly 2 points
(cuts through). CD
Tangent: Line, segment or ray that intersects the circle in
exactly 1 point (touches)
YA or YA
Always perpendicular to radius with endpoint at point of
tangency. XA  YA
C.
Y .
.D
. X
A .
Arcs –
minor arc (less than ½ the circle) AB MB YXA
major arc (at least half the circle) BMY BAM XYB
Sector - area formed by two radii and the
arc formed by them (green area) – like a slice of pizza.
M
C
E
Y
B
X
A
ANGLES in a circle
Central angle: center of circle as vertex and radii as sides.  ACB
Inscribed angle: vertex point on the circle and chords as sides.  AMB
Exterior angle: vertex outside the circle, either secants or tangents as
sides.  E or  AEB
Interior angle: vertex is intersection of two chords inside the circle (not the
center).  AWB
m  ACB = m AB
m  AMB = ½ m AB
m  AEB = ½(m AB – m XY)
m  AWB = ½ (m AB + m MY)
M
C
E
Y
W
B
X
A
SEGMENT Lengths in a circle
Exterior angle forms inverse proportions (* NOTE ORDER)
PR =
PT
PS
PQ
PR • PQ = PT • PS
P
S
T
Q
R
SEGMENT Lengths in a circle
related to chords, secants and tangents
Exterior angles form proportions (* NOTE ORDER)
FG
FJ
=
FG • FH = FJ • FJ
FJ
FH
F
G
J
H
SEGMENT Lengths in a circle
related to chords, secants and tangents
Interior angles form proportions (* NOTE ORDER)
BE
EC
=
BE • ED = AE • BC
AE
ED
A
D
E
B
C
Postulates and theorems about
circles
• The measure of an arc formed by two
adjacent arcs is the sum of the measures of
the two arcs.
• In the same or congruent circles, two minor
arcs are congruent if and only if their
corresponding chords are congruent.
Postulates and theorems about
circles
• If a diameter of a circle is perpendicular to a
chord, then the diameter bisects the chord and its
arc.
• If one chord is perpendicular bisector of another
chord then the first chord is a diameter.
• In the same or congruent circles, two chords are
congruent if and only if they are equidistant from
the center.
Postulates and theorems about
circles
• If an angle is inscribed in a circle, then its measure
is half the measure of its intercepted arc.
• If two inscribed angles of a circle intercept the
same arc, then the angles are congruent.
• A quadrilateral can be inscribed in a circle if and
only if its opposite angles are supplementary.
Postulates and theorems about
circles
• 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.
• If two chords intersect in the interior of a circle,
then the measure of each angle is one half the sum
of the measures of the arcs intercepted by the
angle and its vertical angle.
• If a tangent and a secant, two tangents, or two
secants intersect in the exterior of a circle, then the
measure of the angle formed is one half the
difference of the measures of the intercepted arcs.
Postulates and theorems about
circles
• If two chords intersect in the interior of a circle
then the product of the lengths of the segments of
one chord is equal to the p;product of the lengths
of the segments of the other chord.
• If two secant segments share the same endpoint
outside a circle, then the product of the length of
one secant segment and the length of its external
segment equals the product of the length of the
other secant segment and the length of its external
segment.
Postulates and theorems about
circles
• If a secant segment and a tangent segment
share an endpoint outside a circle, then the
product of the length of the secant segment
and the length of its external segment equals
the square of the length of the tangent
segment.
Postulates and theorems about
circles
• In a circle of radius r, an arc of degree
measure m has arc length equal to (m/360 •
2πr).
• In a circle of radius r, where a sector has an
arc degree measure of m, the area of the
sector is (m/360 • πr2)
• The area of a circle is πr2
• The circumference of a circle is 2πr