1. **State the problem:** We have an isosceles right triangle (45°-45°-90°) with hypotenuse length 38 inches and need to find the length of side $m$. 2. **Recall the properties of a 45°-45°-90° triangle:** The sides are in the ratio $1:1:\sqrt{2}$, where the hypotenuse is $\sqrt{2}$ times the length of each leg. 3. **Set up the formula:** If each leg is $m$, then hypotenuse $= m\sqrt{2}$. 4. **Write the equation:** $$ 38 = m\sqrt{2} $$ 5. **Solve for $m$:** $$ m = \frac{38}{\sqrt{2}} $$ 6. **Rationalize the denominator:** $$ m = \frac{38}{\sqrt{2}} \times \frac{\sqrt{2}}{\sqrt{2}} = \frac{38\sqrt{2}}{2} $$ 7. **Simplify the fraction:** $$ m = 19\sqrt{2} $$ **Final answer:** $m = 19\sqrt{2}$ inches.