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Contents
Symmetrical Properties
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Symmetrical Properties of Circle
Property 1: A circle is symmetrical about every diameter. Hence any chord AB perpendicular to a diameter is bisected by the diameter. Also, any chord bisected by a diameter is perpendicular to the diameter.
Proof: Given a circle, centre O and a chord, AB, with a mid-point D, we are required to show that OĈB = 90°. Join OA and OB. In triangle OAC and OBC, OA
= OB (radii of circle) AC =
BC (given)
OC is
common.
Triangle
OCD is congruent to triangle OBC (SSS property)
OĈA
= OĈB.
Since these are adjacent
angles on a straight line, OĈA = OĈB = 90°
Property 2
In equal
circles or in the same circle, equal chords are equidistant from the centre.
Chords which are equidistant from the centre are equal.
Proof
In the
figure, triangle OAB is rotated through an angle AOA' to triangle
OA'B' about
O.
Since
rotation preserves shape and size, AB = A'B' and OG = OH.
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