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.
(N/A) Let $AB$ and $CD$ be two equal chords of a circle with centre $O$,intersecting at point $E$ inside the circle.
We need to prove that $\angle OEA = \angle OED$.
Draw $OM \perp AB$ and $ON \perp CD$.
In $\Delta OME$ and $\Delta ONE$:
$1$. $OM = ON$ (Equal chords are equidistant from the centre).
$2$. $OE = OE$ (Common side).
$3$. $\angle OME = \angle ONE = 90^\circ$ (By construction).
Therefore,by $RHS$ congruence criterion,$\Delta OME \cong \Delta ONE$.
By $CPCT$,$\angle OEM = \angle OEN$.
Since $M$ lies on $AB$ and $N$ lies on $CD$,this implies $\angle OEA = \angle OED$.
We need to prove that $\angle OEA = \angle OED$.
Draw $OM \perp AB$ and $ON \perp CD$.
In $\Delta OME$ and $\Delta ONE$:
$1$. $OM = ON$ (Equal chords are equidistant from the centre).
$2$. $OE = OE$ (Common side).
$3$. $\angle OME = \angle ONE = 90^\circ$ (By construction).
Therefore,by $RHS$ congruence criterion,$\Delta OME \cong \Delta ONE$.
By $CPCT$,$\angle OEM = \angle OEN$.
Since $M$ lies on $AB$ and $N$ lies on $CD$,this implies $\angle OEA = \angle OED$.
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Similar Questions
State whether the following statements are True or False. Give reasons for your answers.
$(i)$ The line segment joining the centre to any point on the circle is a radius of the circle.
$(ii)$ $A$ circle has only a finite number of equal chords.
$(iii)$ If a circle is divided into three equal arcs,each is a major arc.
$(iv)$ $A$ chord of a circle,which is twice as long as its radius,is a diameter of the circle.
$(v)$ $A$ sector is the region between the chord and its corresponding arc.
$(vi)$ $A$ circle is a plane figure.
$(i)$ The line segment joining the centre to any point on the circle is a radius of the circle.
$(ii)$ $A$ circle has only a finite number of equal chords.
$(iii)$ If a circle is divided into three equal arcs,each is a major arc.
$(iv)$ $A$ chord of a circle,which is twice as long as its radius,is a diameter of the circle.
$(v)$ $A$ sector is the region between the chord and its corresponding arc.
$(vi)$ $A$ circle is a plane figure.
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