Question
If two equal chords of a circle intersect within the circle, prove that the line joining the point of intersection to the centre makes equal angles with the chords.
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Answer
Given: Two equal chords $A B$ and $C D$ of a circle with centre $O$ intersect within the circle. Their point of intersection is $E.$
To prove $\angle O E A=\angle O E D$
Construction: Join $OA$ and $OD$
Proof: In $\triangle O E A$ and $\triangle O E D$
$OE = OE \mid $ Common
$O A=O D \mid$ Radii of a circle
$A E=D E$
$\therefore \triangle O E A \cong \triangle O E D[$ SSS Rule]
$\therefore \angle O E A=\angle O E D$ [c.p.c.t]
To prove $\angle O E A=\angle O E D$
Construction: Join $OA$ and $OD$
Proof: In $\triangle O E A$ and $\triangle O E D$
$OE = OE \mid $ Common
$O A=O D \mid$ Radii of a circle
$A E=D E$
$\therefore \triangle O E A \cong \triangle O E D[$ SSS Rule]
$\therefore \angle O E A=\angle O E D$ [c.p.c.t]
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