1. Circle

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Circle

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Definition
"The set of all points equidistant
from the center".
The locus of all points a fixed
distance from a given (center) point.

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Terms related to a circle
Center - a point
inside the circle
from which all
points on the circle
are equidistant
Radius - the
distance from the
center to any point
on the circle.

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Terms related to the circle
Chord - a line
segment joining any
two points on a
circle.
Diameter - a chord
passing through the
center.

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Terms related to the circle
•Tangent - A line
passing a circle and
touching it at just
one point.
•Secant - A line that
intersects a circle at
two points.

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Terms related to a circle
ARC – a subset of
a circle
Minor Arc < 180°
Major Arc > 180°
Semicircle = 180°

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Terms related to a circle
Central Angle –
angle made by two
radii
Inscribed Angle –
angle made by two
chords intersecting
ON a circle

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Terms related to a circle
•Circumference - the
distance around the
circle. Strictly
speaking a circle is
a line, and so has no
area.
•What is usually
meant is the area of
the region enclosed
by the circle.

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1 Kings 7:23-24 (New
American Standard Bible)
 23
Now he made the sea of cast
metal ten cubits from brim to
brim, circular in form, and its
height was five cubits, and thirty
cubits in circumference.
 24
Under its brim gourds went
around encircling it ten to a
cubit, completely surrounding
the sea; the gourds were in two
rows, cast with the rest.
Even the bible had an approximation for pi! =)

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Terms related to a circle
Segment of a circle
– region on the
circle bounded by a
chord and an arc
Sector – region
bounded by a
central angle and an
arc

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This sector is
sooooooo delicious!

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MK
RK
P
PR
∠RKM
∠RPK
R K
P
M
Given circle P below, what
name applies to each of the
following?

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R K
P
M
Given circle P below, what name applies
to each of the following?

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Indicate whether each statement is TRUE or FALSE.
Every diameter of a circle is a secant of the
circle.
Every radius of a circle is a chord of the
circle.
Every chord of a circle contains exactly two
points of the circle.
If a radius bisects a chord of a circle, then it is
⊥ to the chord.

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Indicate whether each statement is TRUE or FALSE.
The intersection of a line with a circle may be
empty.
A line may intersect a circle in exactly one
point.
The secant which is a perpendicular bisector
of a chord of a circle contains the center of the
circle.

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23. Insights on the Shortest Distance
TSDB2P = The Shortest Distance
Between 2Points
In geometry, TSDB2P is a straight
line.
In sickness, TSDB2P is relief.

24. Insights on the Shortest Distance
TSDB2P = The Shortest Distance Between 2Points
In deep poverty, TSDB2P is realizing you
have plenty to give.
In a career, TSDB2P is integrity.
In parenting, TSDB2P is allowing them to
grow from their own mistakes.

25. Insights on the Shortest Distance
TSDB2P = The Shortest Distance Between 2Points
In friendship, TSDB2P is trust.
In learning, TSDB2P is a mind awaiting
discovery.
In personal growth, TSDB2P is learning
your lesson the first time.

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Inscribed Angle
Theorem
The measure of an
inscribed angle is
equal to one-half
the degree measure
of its intercepted
arc.

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Central Angle
Theorem
The central angle is
twice the inscribed
angle.

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Intersecting Chord
Theorem
When two chords
intersect each other
inside a circle, the
products of their
segments are
equal.
B
A C
D
E

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Angle Formed by
Intersecting
Chords
m∠ABC =
____________________
B
A C
D
E

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Intersecting
Secants Theorem
When two secant
lines intersect each
other outside a
circle, the products
of their segments
are equal.
A
B
C
D
E

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Angle Formed by
Intersecting
Secants
m∠ACE =
________________
A
B
C
D
E

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Intersecting
Tangents
Theorem
When two
SEGMENTS are
tangent to a circle,
the segments are
CONGRUENT.
A
B
C

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Angle Formed by
Intersecting
Tangents
m∠BCA =
______________
A
B
C
D

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Intersecting
Tangent & Secant
Theorem
When two secant
lines intersect each
other outside a
circle, the products
of their segments
are equal.
A
B
C
D

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In the figure, M is the center of the circle.
Name a central angle.
Name a chord that is not
a diameter
Name a major arc.
If m∠KMI = 170°, what
is mKGI ?
What is m∠KHI ?
M
K
G
I H

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Problem: Find the value of x.
Given: Segment AB is tangent to circle C at B.

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Find the measure of each arc/angle:
arc QSR ∠Q ∠R

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Name the arc/s intercepted by:
∠x ∠y ∠z
x
y
z
R
Q
S
P
T
V
A

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Sometimes, secants intersect outside of circles. When
this happens, the measure of the angle formed is equal
to one-half the difference of the degree measures of the
intercepted arcs.
Find the measure of angle 1.
Given: Arc AB = 60o
Arc CD = 100o

43. Hmmm….Prove it first!

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Commonly Used Reasons
Radii of the same circle are congruent.
Base angles of an isosceles triangle are
congruent.
The exterior angle is equal to the sum of the
remote interior angles.
The degree measure of a minor arc is equal to
the measure of the central angle which
intercepts the arc.

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If an angle is inscribed in a circle, then the
measure of the angle equals one-half the
measure of its intercepted arc. (page 170)
P
x
R
Q
S

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PROOF:
STATEMENTS REASONS
Given
Radii of the same circle
are congruent
3. ∆QRS is an isosceles ∆ Def. of an isosceles ∆
4. ∠PQR ≅ ∠QRS Base angles of an
isosceles ∆ are ≅

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PROOF:
STATEMENTS REASONS
5. m∠PQR = m∠QRS Def. of ≅ angles
6. m∠QRS = x Transitive Property
7. m∠PSR = 2x Central Angle Thm.
The minor arc ….

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PROOF:
STATEMENTS REASONS
Transitive Property
Substitution
Division Property of
Equality

49. Let’s learn & relearn...

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The ⊥ line from the center of a circle to a
chord ___________ the chord.bisects

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The measure
of the central
angle is
_________
the measure
of the
inscribed
angle
subtended by
the same arc.
twice

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The
inscribed
angles
subtended
by the
same arc
are
_________.congruent

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A tangent to a circle is ________________
to the radius at the point of tangency.
perpendicular

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Properties of Tangents
Tangent segments to a
circle’s circumference
from any external point
are _______________.
The angle between the
tangent and the chord is
____________________
the intercepted arc.
half of the measure of
congruent
The angle between the
tangent and the chord is
_________________the
inscribed angle on the
opposite side of the
chord.
equal to

55. Additional Exercises

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Find the value of x.
X
4
83

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2. Determine the length AC.

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3. Given a circle with the center O, and OF ⊥ CD
and DE = 20, OF = DF, find OF, EO, DF and DC.

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Find the length AC.

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Given that ED is tangent at C.