To find the measurement with two secants, which expression is correct?

To determine the correct expression for the measurement involving two secants, it is essential to understand the secant-secant product theorem. This theorem states that if two secants are drawn from an external point to a circle, then the products of the lengths of the entire secant and its external segment are equal.

In this case, let's define x as the length of the external segment of one secant, n as the length of the part of the secant that intersects the circle, y as the length of the external segment of the other secant, and m as the length of the part of that secant which intersects the circle. According to the theorem, the relationship can be modeled as:

(x)(x+n) = (y)(y+m)

This reflects that the product of the lengths of the entire secant (which is x + n) and its external part (x) equals the product of the lengths of the entire second secant (y + m) and its external segment (y). Therefore, the equation correctly represents the relationship between the two secants.

The other options do not accurately convey the necessary relationship dictated by the secant-secant theorem. For instance, the option that shows them as mere sums or equal ratios