In this lesson you’ll learn the basics and more details about some other parts of the
circles that lie on or inside the circles but with special names, say an arc- major
and minor, chords, tangents, sector, segment, and secant. It will help to
understand deeper the relationship between- diameter and a chord when they
are perpendicular to each other, the measures of an angle and its minor/ major
arc, and measurement of an angle by two intersecting chords etc. In turn
you’ll be able to solve the real world applications using the learning and
developed skills. Included also are the relevant problems to support
explanation. The contents will be presented by the instructor in own
handwriting with the help of several examples with solutions and using watch video.
(More text below video...)
(Continued from above)Arcs:
An arc is a part of the circumference of a circle and defined as: 'an arc of a circle is
the part of the circle between two points on the circle'. The longer arc is called the
major arc while the shorter one is called the minor arc. Arc is measured in degrees and
length. If the measure of minor arc is i.e. the measure of the central angle intercepted
by the minor arc, then the measure of major arc is
(360
-)
i.e. the measure of the central angle intercepted by the minor arc, then the measure
of major arc is (360
-
). E.g. if measure of a minor arc is 100,
then major arc is (360
- 100)
= 260.
Now look at the relationship of radius and measure of central angle: if r is the
radius of the circle and is the measure of central angle, then length of the arc
intercepted by the angle
is the measure of central angle, then length of the arc intercepted by the angle
is given
by
/ 360
* 2r.
E.g. if
= 120,
then length of
the arc intercepted is 4/3
units. Notice that an arc is measured commonly not
by its length, but more often expressed as measure of the angle whose vertex is
the center of the circle and its rays intercept the endpoints of the arc. Hence
an arc can be anywhere from 0
to 360.
Types of Arcs:
major arc, minor arc, and semicircle
• If mAOB
< 180,
points A and B and the points of circle in
the interior of AOB
make up minor arc AB, written as
• Points A and B and the points of the circle not in the interior
of AOB
make up major arc AB, written as
or
• If mAOC
= 180,
points A and C separate circle O into two equal
parts, each of which is called a semicircle. In (Fig: 1),
and
two different semicircles
Congruent Arcs:
Central angle is an angle whose vertex is the center of a circle ‘O’.
Any central angle intercepts the circle at two points, thus defines the arc (Fig: 2).
Congruent arcs are arcs of the same circle or of congruent circles that are
equal in measure. In (Fig: 3, above), if OO'
and mCD = mC'D' = 60,
then
Arc Addition Postulate
If AB and BC are two arcs of the same circle having a common endpoint and
no other points in common, then AB + BC
= and mAB + mBC
= m.
Theorems
•
In a circle or in congruent circles, if central angles are congruent, then their intercepted arcs are congruent (Figure below).
if OO',
AOBCOD,
and AOBA'O'B'.
then
and
•
In a circle or in congruent circles, central angles are congruent if their intercepted arcs are congruent.
•
In a circle or in congruent circles, central angles are congruent if and only if their intercepted arcs are congruent.
E.g. (In figure below) P, Q, S,
and R are points on circle O, mPOQ
= 100,mQOS
= 110,
and mSOR
= 35.
Find mROP.
mROP
will be- [360
– (100
+ 110
+ 35)]
= 115,
as the final answer.
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-
)
i.e. the measure of the central angle intercepted by the minor arc, then the measure
of major arc is (360
r.
E.g. if
AOB
< 180
or 
and
two different semicircles 
O
and mAB + mBC
= m
