If two equal chords of a circle intersect within the circle, prove that the segments of one chord are equal to corresponding segments of the other chord.
Solution:
Let’s suppose that we have circle with centre O. There are two equal chords AB and CD intersecting at point E.
Construction: Draw
and
. Join OE.
In
and
, we have
(Each equal to
)
(Common)
(Equal chords are equidistant from the centre)
Therefore, by RHS congruence rule, we have
(Corresponding parts of congruent triangles are equal) (1)
We also have
(given) (2)
(3)
(Perpendicular from the centre of the circle to a chord bisects the chord)
Adding (1) and (3) we get,
(4)
Subtracting (4) from (2), we get
(5)
From (4) and (5), we can say that if two equal chords of a circle intersect within the circle, then the segments of one chord are equal to corresponding segments of the other chord.
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