If PA and PB are tangents from an outside point P. such that PA = 10 cm and ∠APB = 60°. Find the length of chord AB.

Home » If PA and PB are tangents from an outside point P. such that PA = 10 cm and ∠APB = 60°. Find the length of chord AB.

If PA and PB are tangents from an outside point P. such that PA = 10 cm and ∠APB = 60°. Find the length of chord AB.

Given,

$AP=10cm$

$\angle APB={{60}^{\circ }}$

According to the figure

We know that,

A line drawn from centre to point from where external tangents are drawn, bisects the angle made by tangents at that particular point

$\angle OPB=\angle APO=\frac{1}{2}\times {{60}^{\circ }}={{30}^{\circ }}$

And, bisected perpendicularly the chord AB

$\therefore AB=2AM$

In $\vartriangle AMP$ ,

$\sin {{30}^{\circ }}=\frac{\text{Opposite}}{\text{Hypotenuse}}=\frac{AM}{AP}$

$AM=AP\sin {{30}^{\circ }}$

$\frac{AP}{2}=\frac{10}{2}$

$=5cm$ [As AB = 2AM]

So, $AP=2AM=10cm$

And, $AB=2AM=10cm$

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