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

Question

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

Team Careers360 · Accepted Answer

Correct Answer: 6 cm

Solution :
Given, $OP = 12$ cm and $\angle APB=120^\circ$
In $ riangle OPA$ and $ riangle 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, $ riangle OPA$ $\cong$ $ riangle OPB$
So, $\angle APO=\frac{120^\circ}{2}=60^\circ$
Now, in $ riangle 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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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∘?Option 1: 24 cmOption 2: 6 cmOption 3: 8 cmOption 4: cannot be determined

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