Question

which equation results from applying the secant and tangent segment theorem to this figure?

x(x + 2) = (x + 4)
x(x + 4) = (x + 2)
x(x + 4) = (x + 2)^2
x(2x + 4) = (x + 2)^2

Explanation:

Step1: Recall the secant - tangent segment theorem

If a tangent segment and a secant segment are drawn to a circle from an external point, then the square of the length of the tangent segment is equal to the product of the lengths of the entire secant segment and its external part.
Let the tangent segment length be \(l_{t}=x + 2\), the external part of the secant segment be \(l_{e}=x\), and the entire secant segment length be \(l_{s}=x+(x + 4)=2x+4\). According to the theorem \(l_{t}^{2}=l_{e}\times l_{s}\).

Step2: Substitute the values into the formula

Substitute \(l_{t}=x + 2\), \(l_{e}=x\), and \(l_{s}=2x + 4\) into the formula \((x + 2)^{2}=x\times(2x + 4)\)

Answer:

\(x(2x + 4)=(x + 2)^{2}\)