Step-by-step explanation:
GIVEN :
A circle with centre o. TP and TQ are two tangents at P and Q.
TO PROVE :
ANGLE APB=2ANGLE OAB
PROOF :
TP =TQ (Tangents from an external point)
PTQ is an isosceles triangle
In ∆ PTQ
ANGLE (PTQ)+ ANGLE (TPQ)+ANGLE (TQR)=180°( Angle sum property of a triangle )
ANGLE (PTQ)+2ANGLE(TPQ)=180°
2ANGLE(TPQ)=180°-ANGLE(PTQ)
ANGLE(TPQ)=90°-ANGLE(PTQ) ( i )
ANGLE (OPT )=90° ( Radius is perpendicular to the tangent )
ANGLE (OPT)= ANGLE (OPQ)+ANGLE(TPQ)
90°-ANGLE(TPQ)=ANGLE (OPQ)
SUBSTITUTE EQUATION ( i )
90°-90°+PTQ/2=ANGLE (OPQ)
PTQ/2=ANGLE (OPQ)
ANGLE (PTQ)= 2ANGLE OAB
HENCE PROVED.....
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Verified answer
Step-by-step explanation:
GIVEN :
A circle with centre o. TP and TQ are two tangents at P and Q.
TO PROVE :
ANGLE APB=2ANGLE OAB
PROOF :
TP =TQ (Tangents from an external point)
PTQ is an isosceles triangle
In ∆ PTQ
ANGLE (PTQ)+ ANGLE (TPQ)+ANGLE (TQR)=180°( Angle sum property of a triangle )
ANGLE (PTQ)+2ANGLE(TPQ)=180°
2ANGLE(TPQ)=180°-ANGLE(PTQ)
ANGLE(TPQ)=90°-ANGLE(PTQ) ( i )
ANGLE (OPT )=90° ( Radius is perpendicular to the tangent )
ANGLE (OPT)= ANGLE (OPQ)+ANGLE(TPQ)
90°-ANGLE(TPQ)=ANGLE (OPQ)
SUBSTITUTE EQUATION ( i )
90°-90°+PTQ/2=ANGLE (OPQ)
PTQ/2=ANGLE (OPQ)
ANGLE (PTQ)= 2ANGLE OAB
HENCE PROVED.....