Question

Two chords intersect at a point within the circle and the diameter through this point bisects the angle between the chords. Then

• A. one of the chords is twice the length of the other.
• B. one of the chords is four times the length of the other.
• C. two chords have the same length.
• D. one of the chords is one third the length of the other.

Answer 1

The correct option is C

two chords have the same length.

Two chords AB and CD intersect at point P within the circle of centre O.

Draw $OM\perp AB$ and $ON\perp CD$ which are perpendicular bisectors of chord AB and CD respectively.

Given, diameter RS through point P bisects $\angle BPD$.

$⟹\angle BPO=\angle DPO$ - - - - - (i)

Now, in $△OMP$ and $△ONP$,

$\angle OPM=\angle OPN$ (from (i))

$\angle OMP=\angle ONP$ (each 90°)

OP = OP (common)

$\therefore △AMP\cong △ANP$ [by AAS congruence rule]

$⟹OM=ON$ [c.p.c.t.]

$⟹AB=CD$ [Chords equidistant from the centre are equal]

Hence both the chords have the same length.