Mathematics
In the given figure, two chords AC and BD of a circle intersect at E. If arc AB = CD, prove that : BE = EC and AE = ED.
Answer
Join AB and CD.
Given, arc AB = arc CD.
Since, equal arcs subtend equal chords.
⇒ chord AB = chord CD
In △ AEB and △ DEC,
⇒ AB = CD (Proved above)
⇒ ∠BAE = ∠CDE (Angles in the same segment subtended by arc BC)
⇒ ∠ABE = ∠DCE (Angles in the same segment subtended by arc AD).
By ASA Congruence,
∴ △ AEB ≅ △ DEC
Since corresponding parts of congruent triangles are equal.
⇒ BE = EC and AE = ED.
Hence, proved that BE = EC and AE = ED.
Related Questions
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If P is a point on a circle with centre O. If P is equidistant from the two radii OA and OB, prove that arc AP = arc PB.
In the given figure, two chords AB and CD of a circle intersect at a point P.
If AB = CD, prove that: arc AD = arc CB.
If two sides of a cyclic quadrilateral are parallel, prove that:
(i) its other two sides are equal,
(ii) its diagonals are equal.