Question

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

The secant - tangent segment theorem states that if a tangent segment and a secant segment are drawn from an external point to a circle, 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 length of the external part of the secant be \( x \), the length of the entire secant be \( x+(x + 4)=2x + 4\) (since the internal part of the secant is \( x + 4\) and the external part is \( x\)), and the length of the tangent be \( x+2\).

Step2: Apply the theorem

According to the secant - tangent theorem, \( (\text{length of tangent})^2=\text{length of external part of secant}\times\text{length of entire secant}\).

Substituting the values, we get \((x + 2)^2=x\times(2x + 4)\).

Answer:

\( x(2x + 4)=(x + 2)^2\) (the fourth option)