1. **Problem statement:** We have a right triangle with one leg of length 7, an angle of 45° adjacent to the unknown side $a$, and the hypotenuse $b$. We need to find the length of side $a$. 2. **Relevant formula:** In a right triangle, the side adjacent to an angle $\theta$ relates to the hypotenuse by the cosine function: $$\cos(\theta) = \frac{\text{adjacent side}}{\text{hypotenuse}}$$ 3. **Important note:** Here, side $a$ is adjacent to the 45° angle, and side 7 is opposite that angle. We can also use the tangent function: $$\tan(\theta) = \frac{\text{opposite side}}{\text{adjacent side}}$$ 4. **Apply tangent:** Using $\theta = 45^\circ$, opposite side = 7, adjacent side = $a$: $$\tan(45^\circ) = \frac{7}{a}$$ 5. **Evaluate tangent:** Since $\tan(45^\circ) = 1$, we have: $$1 = \frac{7}{a}$$ 6. **Solve for $a$:** Multiply both sides by $a$ and divide by 1: $$a = 7$$ 7. **Answer:** The length of side $a$ is 7 units.