1. **State the problem:** We have two secants, HIJ and LKJ, intersecting outside the circle at point J. Given lengths are $JI=31$, $JK=32$, and $KL=16$. We need to find the length of $HI$. 2. **Formula used:** For two secants intersecting outside a circle, the product of the external segment and the entire secant length is equal for both secants: $$JI \times JH = JK \times JL$$ where $JH = JI + HI$ and $JL = JK + KL$. 3. **Calculate $JL$:** $$JL = JK + KL = 32 + 16 = 48$$ 4. **Set up the equation:** $$JI \times JH = JK \times JL$$ Substitute known values: $$31 \times (31 + HI) = 32 \times 48$$ 5. **Simplify the right side:** $$32 \times 48 = 1536$$ 6. **Expand the left side:** $$31 \times 31 + 31 \times HI = 1536$$ $$961 + 31HI = 1536$$ 7. **Isolate $HI$:** $$31HI = 1536 - 961$$ $$31HI = 575$$ 8. **Divide both sides by 31:** $$HI = \frac{575}{31}$$ Show cancellation: $$HI = \frac{\cancel{575}}{\cancel{31}}$$ 9. **Calculate the division:** $$HI \approx 18.548$$ 10. **Round to the nearest tenth:** $$HI \approx 18.5$$ **Final answer:** The length of $HI$ is approximately **18.5** units.