Question

In the given figure XY and X'y are parallel tangents
to O centered circle and AB is another tangent on the
point C then prove that angleAOB = 90°​

Answer 1

In ΔOPAandΔOCA

OP=OC (Radii of the same circle)

AP=AC (Tangent from point A)

AO=AO (Common side)

ΔOPA≅ΔOCA (SSS congruence criterion)

Therefore, P is congruent to C,  and O are same

∠POA=∠COA    (1)

Similarly,  

∠QOB≅∠OCB

∠QOB=∠COB      (2)

As POQ is the diameter of the circle, it is a straight line.

So  ∠POA+∠COA+∠COB+∠QOB=180

 

From equation (1) and equation (2)

2∠COA+2∠COB=180  

∠COA+∠COB=90  

∠AOB=90

Hence proved

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