What is the formula for the ratio between a tangent line and the segments of a secant line?

The formula for the relationship between a tangent line and a secant line can be understood through the context of a circle and the properties of tangents and secants. When you have a tangent line that touches a circle at one point and a secant line that intersects the circle at two points, there is a specific relationship given by the power of a point theorem.

According to this theorem, the square of the length of the tangent segment drawn 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 farthest point on the circle) and the external segment of the secant (from the external point to the nearest point of intersection with the circle). This leads to the equation where the length of the tangent squared equals the product of the external segment of the secant and the whole secant segment, which is precisely represented as:

tangent² = external secant segment * whole secant.

This relationship demonstrates how geometric properties of circles dictate the distances involved and emphasizes the connection between tangent lines and secant lines in a clearly defined mathematical expression.