Question : Two equal tangents PA and PB are drawn from an external point P on a circle with centre O. What is the length of each tangent, if P is 12 cm from the centres and the angle between the tangents is 120$^\circ$?
Option 1: 24 cm
Option 2: 6 cm
Option 3: 8 cm
Option 4: cannot be determined
Correct Answer: 6 cm
Solution :
Given, $OP = 12$ cm and $\angle APB=120^\circ$
In $\triangle OPA$ and $\triangle OPB$,
$PA=PB$ (equal tangents)
$\angle OAP=\angle OBP=90^\circ$ (tangent is perpendicular to the radius at its point of contact)
$OP=OP$ (common side)
So, $\triangle OPA$ $\cong$ $\triangle OPB$
So, $\angle APO=\frac{120^\circ}{2}=60^\circ$
Now, in $\triangle APO$, $\cos 60^\circ = \frac{PA}{PO}$
⇒ $\frac{1}{2}=\frac{PA}{12}$
⇒ $PA$ = 6 cm
Hence, the correct answer is 6 cm.
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