1. **State the problem:** We have a right triangle with vertices I, J, and H. The right angle is at vertex I. The angle at vertex J is 54° and the side JH (hypotenuse) has length $\sqrt{71}$. We need to find the length of side HI. 2. **Identify the sides:** Since the right angle is at I, side JH is the hypotenuse. Side HI is adjacent to angle J, and side IJ is opposite angle J. 3. **Use trigonometry:** The cosine of angle J relates the adjacent side HI to the hypotenuse JH: $$\cos(54^\circ) = \frac{HI}{\sqrt{71}}$$ 4. **Solve for HI:** $$HI = \sqrt{71} \times \cos(54^\circ)$$ 5. **Calculate the value:** $$\cos(54^\circ) \approx 0.5878$$ $$HI \approx \sqrt{71} \times 0.5878 \approx 8.4261 \times 0.5878 \approx 4.95$$ 6. **Round the answer:** $HI \approx 5.0$ (rounded to the nearest tenth) **Final answer:** $$\boxed{5.0}$$