A chord is a straight line joining 2 points on the circumference of a circle.
Theorem: A radius or diameter that is perpendicular to a chord divides the chord into two equal parts and vice versa.
In the above circle, if the radius OB is perpendicular to the chord PQ then PA = AQ.
Converse: The perpendicular bisector of a chord passes through the center of a circle.
In the above circle, OA is the perpendicular bisector of the chord PQ and it passes through the center of the circle. OB is the perpendicular bisector of the chord RS and it passes through the center of the circle.
We can use this property to find the center of any given circle.
Example:Determine the center of the following circle.
Step 1: Draw 2 non-parallel chords
Step 2: Construct perpendicular bisectors for both the chords. The center of the circle is the point of intersection of the perpendicular bisectors.
Theorem: Congruent Chords are equidistant from the center of a circle.
Converse: Chords equidistant from the center of a circle are congruent.
If PQ = RS then OA = OB or
If OA = OB then PQ = RS
Theorem: If two chords in a circle are congruent then their intercepted arcs are congruent.
Converse: If two arcs are congruent then their corresponding chords are congruent.Theorem on chords and arcs and shows an example on how to use theorem
The figure is a circle with center O. Given PQ = 12 cm. Find the length of PA.
The radius OB is perpendicular to PQ. So, OB is a perpendicular bisector of PQ.
The figure is a circle with center O and diameter 10 cm. PQ = 1 cm. Find the length of RS.
OP = OQ – PQ
= 5 cm – 1 cm = 4 cm
Using Pythagoras’ theorem,
Since OQ is a radius that is perpendicular to the chord RS, it divides the chord into two equal parts.
RS = 2RP = 2 × 3 = 6 cmIf a diameter is perpendicular to a chord, then it bisects the chord and its arc.
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