Chord:
A chord of a circle is a line segment whose endpoints are points
on the circle. E.g. in the Fig: 4, AB and AOC are chords of circle O.
Thus, a diameter is a special chord of a circle that has the center
of the circle as one of its points. In the Fig: 4, AOC is the diameter.
(More text below video...)
Important Properties of chords
Look at the important properties of chords (Fig: 5):
•
every chord defines an arc whose endpoints are the same as those of the chord and AD = DB.
E.g. diameter and semicircle are the chord and arc that share the same endpoints.
•
the central angle forms an isosceles triangle, with chord as one side and the other
two sides are rays that make the central angle.
•
the only diameter perpendicular to the given chord, is the perpendicular bisector
also of that chord.
Chord Theorems: •
in a circle or in congruent circles, congruent central angles have congruent chords.
• in a circle or in congruent circles, congruent chords have congruent central angles (Fig: 6).
•
in a circle or in congruent circles, two chords are congruent if
and only if their central angles are congruent.
•
In a circle or in congruent circles, congruent arcs have congruent chords (Fig: 7)
• in a circle or in congruent circles, congruent chords have congruent arcs (Fig: 8).
• in a circle or in congruent circles, two chords are congruent if and only if their arcs are congruent.
Chords Equidistant from the Center of a Circle
Theorem- A diameter perpendicular to a chord bisects the chord and its arcs.
Given: Diameter
of circle O, chord at E.
Prove , and
Proof: Draw AEO
= BEO
= 90 (Perpendicular lines intersect to form right angle)
=>
AOEBOE
(by HL)
=>
and
AOEBOE
=>
(In a circle, congruent central angles have congruent arcs.)
=>
AOD
BOD
(Supplements of congruent angles are congruent.)
=>
Corollary
A line through the center of a circle that is perpendicular to a chord bisects the chord and its arcs.
Apothem
An apothem of a circle is a perpendicular line segment from the center of a circle to
the midpoint of a chord. In the diagram, E is the mid point of chord AB in circle O, ABCD, and OE (Fig: 10). Then OE is the apothem.
Theorems: -
• The perpendicular bisector of the chord of a circle contains
the center of the circle (Fig: 11).
• If two chords of a circle are congruent, then they are
equidistant from the center of the circle i.e.
(Fig: 12)
•
If two chords of a circle are equidistant from the center of the circle, then the chords are congruent i.e.
(Fig: 13)
•
Two chords are equidistant from the center of a circle if and only if the chords are congruent.
•
In a circle, if the lengths of two chords are unequal, then the shorter chord is farther
from the center i.e. (Fig 14, above).
Polygons Inscribed in a Circle
If all of the vertices of a polygon are points of a circle, then the polygon is said to
be inscribed in the circle(Fig: 15, above). In other words it can also be expressed that
the circle is circumscribed about the polygon.
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of circle O, chord 
at E.
, and
AEO
= 
AOE
and
(In a circle, congruent central angles have congruent arcs.)
CD, and OE (Fig: 10). Then OE is the apothem.
(Fig: 12)
(Fig: 13)
(Fig 14, above).