Question

17. In the figure O is the centre of the circle and AC is the diameter. AB and CD are
two parallel chords. Angle C=30°.OQ=3 cm. OP is perpendicular to AB and OQ is
perpendicular to CD.

a) OC =
QC=​

Answer 1

Answer:

AB and AC are two equal chords of a circle with centre O.

OP⊥AB and OQ⊥AC.

To prove: PB=QC

Proof: OP⊥AB

⇒AM=MB .... (perpendicular from centre bisects the chord)....(i)

Similarly, AN=NC....(ii)

But, AB=AC

2AB =2AC⇒MB=NC ...(iii) From (i) and (ii)

Also, OP=OQ (Radii of the circle)

and OM=ON (Equal chords are equidistant from the centre)

⇒OP−OM=OQ−ON

⇒MP=NQ ....(iv) (From figure)

In ΔMPB and ΔNQC, we have

∠PMB=∠QNC (Each =90)

MB=NC ( From (iii) )

MP=NQ ( From (iv) )

∴ΔPMB≅ΔQNC (SAS)

⇒PB=QC (CPCT)