1. **State the problem:** We have a circle with a secant line passing through points H, G, and F, and a tangent line EF touching the circle at point E. We know the lengths HG = 8 and GF = 9, and we need to find the length of the tangent segment EF. 2. **Relevant formula:** The tangent-secant theorem states that 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. Mathematically: $$EF^2 = HG \times (HG + GF)$$ 3. **Apply the values:** $$EF^2 = 8 \times (8 + 9)$$ $$EF^2 = 8 \times 17$$ $$EF^2 = 136$$ 4. **Solve for EF:** $$EF = \sqrt{136}$$ 5. **Calculate the square root:** $$EF \approx 11.6619$$ 6. **Round to the nearest tenth:** $$EF \approx 11.7$$ **Final answer:** The length of the tangent segment EF is approximately 11.7 units.