In these lessons, we will learn theorems that involve chords of a circle.
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.
Determine the centre of the following circle.
Step 1: Draw 2 non-parallel chords
Step 2: Construct perpendicular bisectors for both the chords. The centre 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.This video explains the theorem on chords and arcs and shows an example on how to use theorem. It also shows the perpendicular bisector theorem.
This video discusses the following theorems:
(1) Congruent central angles have congruent chords
(2) Congruent chords have congruent arcs
(3) Congruent arcs have congruent central angles
The figure is a circle with centre 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 centre 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 cm
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