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Prove that the angle between the …

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Prove that the angle between the two tangent drawn from an external point to a circle is supplementary to the angle subtended by the line segment joining the points of contact at the centre

Posted by Prajval Rawat 6 years, 5 months ago

CBSE > Class 10 > Mathematics

1 answers

Sia ? 6 years, 5 months ago

{tex}\angle{/tex}OAP = 90^o ...........(1) [Angle between tangent and radius through the point of contact is 90^o ]
{tex}\angle{/tex}OBP = 90^o ...........(2) [Angle between tangent and radius through the point of contact is 90^o ]
{tex}\therefore{/tex} OAPB is quadrilateral

{tex}\therefore{/tex} {tex}\angle{/tex}APB + {tex}\angle{/tex}AOB + {tex}\angle{/tex}OAP + {tex}\angle{/tex}OBP = 360^o [Angle sum property of a quadrilateral]
{tex}\Rightarrow{/tex} {tex}\angle{/tex}APB + {tex}\angle{/tex}AOB + 90^o + 90^o = 360^o [From (1) and (2)]
{tex}\Rightarrow{/tex} {tex}\angle{/tex}APB + {tex}\angle{/tex}AOB = 180^o
{tex}\Rightarrow{/tex} {tex}\angle{/tex}APB and {tex}\angle{/tex}AOB are supplementary

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