1. **State the problem:** We have a right triangle with sides 9 mm, 15 mm, and an unknown side $b$. 2. **Identify the formula:** For a right triangle, the Pythagorean theorem applies: $$a^2 + b^2 = c^2$$ where $c$ is the hypotenuse (the longest side). 3. **Determine the hypotenuse:** The longest side given is 15 mm, so $c = 15$ mm. 4. **Apply the formula:** Let the unknown side be $b$, and the other known side be 9 mm. $$9^2 + b^2 = 15^2$$ 5. **Calculate squares:** $$81 + b^2 = 225$$ 6. **Isolate $b^2$:** $$b^2 = 225 - 81$$ $$b^2 = 144$$ 7. **Find $b$ by taking the square root:** $$b = \sqrt{144}$$ $$b = 12$$ 8. **Conclusion:** The missing length $b$ is 12 mm.