Question

In the given fi gure, AF, CE and BD are diameters intersecting at point O. If AB = CD, then fi nd the measure

of angle AOB.
​

Answer 1

Answer:

Given: 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

⇒  

2

AB

​  

=  

2

AC

​  

 

⇒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)

Step-by-step explanation:

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