1. **State the problem:** We need to find the measure of segment $GH$ given that $EF = 12$, $FG = 8$, and $GH = -6 + 2x$. 2. **Use the tangent-secant theorem:** When a tangent and a secant are drawn from a point outside a circle, the square of the tangent segment equals the product of the entire secant segment and its external part. 3. **Identify segments:** Here, $EF$ is the tangent segment, so $EF = 12$. The secant segment is $FH = FG + GH = 8 + (-6 + 2x) = 2 + 2x$. The external part of the secant is $FG = 8$. 4. **Apply the theorem:** $$EF^2 = FG \times FH$$ $$12^2 = 8 \times (2 + 2x)$$ $$144 = 8(2 + 2x)$$ 5. **Simplify and solve for $x$:** $$144 = 16 + 16x$$ $$144 - 16 = 16x$$ $$128 = 16x$$ $$x = \frac{128}{16}$$ $$x = 8$$ 6. **Find $GH$:** $$GH = -6 + 2x = -6 + 2(8) = -6 + 16 = 10$$ **Final answer:** $GH = 10$ which corresponds to option D.