When a minor arc and a chord share the same endpoints, we called the arc the ARC OF THE CHORD.

presenterawfulElectronics - Devices

Oct 10, 2013 (3 years and 10 months ago)

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When a minor arc and a chord share the
same endpoints, we called the arc the
ARC
OF THE CHORD
.

A misconception is that a radius is a chord
but it is not beca
use both of its endpoints
are not on the circle.


However, a diameter
of a circle is a chord.



In a circle or in congruent circles, two
minor arcs are cong
ruent if and only if
their corresponding chords are congruent.

(Hint): Are the arcs of
diameters

congruent?.... Yes of course, they are
semicircles...




In
a circle or in congruent circles, two
chords are congruent if and only if they are
equidistant from the center.

Chords are congruent if they are
equidistant from the center, they are also
congruent if there arcs are the same size.



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If two segments from t
he same
exterior point are tangent

to the
c
ircle, then they are congruent.
(

)

(Hint):


To prove this
theorem to be true create
segments AM, and radii CM,
and BM.


Now look for
congruent triangles.


This

should

is
a commonly used theorem for questions.... know it well.

-

If a line is tangent to a circle, then it is
perpendicular to the radius drawn to the point of
tangency.

The Converse states:


In a plane, if a
line is perpendicular to a radius of a circle at the

endpoint on the circle, then the line is a tangent
of the circle.



In a circle, if a diameter is perpendicular to a
chord, then it bisects the chord and its arc.


(Hint): This diagram creates right triangles if
you add radius OA or OB.


Many good
questi
ons center around this theorem... know it
well



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Inscribed Angles:

Inscribed Angles & Intercepted Arcs
-

An
INSCRIBED ANGLE

is an angle whose vertex
is on the circle and whose sides each contain
chords of a circle.

(Hint):
ADB is inscribed on
.


An inscribed
arc has its vertex on the circle.

I
f an angle is
inscri
bed in a circle, then
the
measure of the angle equals one
-
half the
measure of its intercepted arc.


(Hint):


This
relationship shows up everywhere.


It is very
important.


The measure of the arc is always
double its inscribed angle.





Definition of an
INSCRIBED POLYGON

-

A
polygon is an inscribed polygon if each of its
vertices lies on a circle.


The polygon below
would be called an inscribed quadrilateral or
an
other name for it is cyclic quadrilateral.

If a quadrilateral is inscribed
in a circle, then its opposite
angles are supplementary.


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If two inscribed angles of a circle or
congruent circles intercept congruent arcs
or the same arc, then the angles are

congruent.


(Hint): This theorem makes a
lot of sense.


If the inscribed angles are
sitting on the same arc then they are all half
the size of the arc and thus all congruent to
each other.


If an inscribed angle of a circle intercepts a
semicircle, then

the angle is a right angle.


(Hint):


Once again this makes a lot of
sense.


The arc is a semicircle (180) because
AB is a diameter and so then inscribed angle
will be half..... 90.




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A central angle is equal to its
intercepted arc
.


If an angle is inscribed in a circle, then
the measure of the angle equals one
-
half the measure of its intercepted arc.


(Hint):


This relationship shows up
everywhere.


It is very importa
nt.


The
measure of the arc is al
ways double its
inscribed angle,
ie

when the vertex is on
the circumference.




If two secants, a secant and a tangent, or two tangents
intersect in the exterior of a circle, then the measure of the
angle formed is one
-
h
alf the positive difference of the
measures of the intercepted arcs.





(Big Arc
-

Little Arc) divided
by 2


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If a secant and a tangent intersect at the
point of tangency, then the measure of each
angle formed is one
-
half the measure of its
intercepted
arc.


(Hint): This theorem states
that


half of


(green arc) =


ABT.


It would
also be true that half
(black arc) =
ABF


(green arc) =


ABT


(black=
ABF

If an inscribed angle of a circle

intercepts a
semicircle, then the angle is a right angle.


(Hint):


Once again this makes a lot of
sense.


The arc is a semicircle (180) because
AB is a diameter and so then inscribed angle
will be half..... 90.




If two secants in
tersect in the interior of a circle, then the
measure of an angle formed is one
-
half the sum of the
measures of the arcs intercepted by the angle and its vertical
angle.

(Big Arc
+

Little Arc) divided by 2



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Power Theorems

Intersecting Secants Theorem

S
ecant Secant

If two secants, AE and AD, also cut the circle at B and C
respectively, then AE
*

AB = AC
*

AD

Whole x outside = whole x outside

. (Corollary of the tangent
-
secant theorem)

Power Theorems

Theorem of Intersecting Chords


If two chords AB and CD of a circle
intersect at a po
int E inside the
circle, then the product of the
lengths of the segments AE and EB
of the chord AB is equal to the
product of the lengths of the
segments CE and ED of the chord
CD.




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Power Theorems

Tangent

Secant
:

A

secant segment and a tangent segment
share an
endpoint not on the circle. This theorem states that the length of the tangent
segment squared is equal to the product of the secant segment and its
external segment.

Theorem

Tangent

squared = whole times outside

TP
= PA* PB



If a tangent from an external point P meets the circle at
T

and a
secant from the external point
P

meets the circle at
B

and
A
respectively
, then
PT
2

=
PB
×
PA
. (tangent
-
secant theorem)