State whether the following statement is 'True' or 'False' and provide reasons for your answer: The length of a tangent from an external point to a circle is always greater than the radius of the circle.
(B) The statement is 'False'.
Let the radius of the circle be $r$ and the length of the tangent from an external point $P$ be $L$.
In a right-angled triangle formed by the radius, the tangent, and the line segment connecting the center to the external point, the hypotenuse is the distance from the center to the external point $(d)$.
By the Pythagorean theorem, $d^2 = r^2 + L^2$.
Since $d > r$, it follows that $L = \sqrt{d^2 - r^2}$.
Depending on the distance $d$ of the external point from the center, the length of the tangent $L$ can be greater than, equal to, or less than the radius $r$.
For example, if the external point is very close to the circle, the tangent length can be smaller than the radius.
Let the radius of the circle be $r$ and the length of the tangent from an external point $P$ be $L$.
In a right-angled triangle formed by the radius, the tangent, and the line segment connecting the center to the external point, the hypotenuse is the distance from the center to the external point $(d)$.
By the Pythagorean theorem, $d^2 = r^2 + L^2$.
Since $d > r$, it follows that $L = \sqrt{d^2 - r^2}$.
Depending on the distance $d$ of the external point from the center, the length of the tangent $L$ can be greater than, equal to, or less than the radius $r$.
For example, if the external point is very close to the circle, the tangent length can be smaller than the radius.
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