How does the length of a tangent relate to a secant?

The relationship between the length of a tangent drawn from an external point to a circle and a secant line that intersects the circle is described by a specific theorem in geometry. This theorem states that the square of the length of the tangent segment (from the external point to the point of tangency) is equal to the product of the lengths of the entire secant segment (from the external point to the point of intersection with the circle) and the length of the external part of the secant (from the external point to the point where the secant intersects the circle).

In essence, if you denote the length of the tangent as (t), the entire length of the secant as (s), and the external part of the secant as (e), the relationship can be mathematically expressed as:

[ t^2 = e \cdot s ]

This illustrates why the tangent squared equals the product of the lengths of the external segment of the secant and the entire length of the secant. This relationship is crucial in various problems involving circles, as it provides a means to find unknown lengths when certain other lengths are known.

The other options do not accurately represent the geometric relationship defined by this theorem, making the correct