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Prove that the lengths of the tangents drawn from an external point to a circle are equal.
Given: Tangents drawn from an external point to a circle.
To do: To prove that the lengths of both the tangents are equal.
Solution: Let there is a circle with center O. PR and QR are tangents to the circle from an external point R touching the circle at P and Q respectively.
In $\vartriangle $OPR and $\vartriangle $OQR,
OP$=$OQ ...........radius of the circle
Since PR and QR are tangents to the circle and it is known that tangent to a circle always perpendicular to the radius to the point of contact.
$\therefore$$\angle $OPR$=$$\angle
$OQR $=$$90^{o}
$
OR is a common side.
$\therefore$OR$=$OR
$\therefore$ $\vartriangle$ OPR $\cong $ $\vartriangle$OQR ......S.A.S.rule
As known corresponding parts of concurrent triangles are always equal.
$\therefore$PR$=$QR
Hence proved that the lengths of the tangents drawn from an external point to a circle are equal.
Updated on: 10-Oct-2022
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