Question

In the given figure, o is the centre of a circle and AB is a chord. Ox is perpendicular to AB. Prove that X is the mid-point of AB. B A Х​

Answer 1

Step-by-step explanation:

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)