If the angle between two tangents …
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Sia ? 6 years, 6 months ago
We know that tangent is always perpendicular to the radius at the point of contact.
So, ∠OAP = 90
We know that if 2 tangents are drawn from an external point, then they are equally inclined to the line segment joining the centre to that point.
So, ∠OPA = 12∠APB = 12{tex}\times{/tex}60° = 30°
According to the angle sum property of triangle-
In ∆AOP, ∠AOP + ∠OAP + ∠OPA = 180°
{tex}\Rightarrow{/tex} ∠AOP + 90° + 30° = 180°
{tex}\Rightarrow{/tex} ∠AOP = 60°
So, in triangle AOP
tan angle AOP = AP/ OA
{tex}\sqrt 3 = \frac{{AP}}{a}{/tex}
therefore, {tex}AP = \sqrt 3 a{/tex}
hence, proved
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