1. **Problem Statement:** Determine the value of $k$ using the given diagram (top-right coordinate graph with angle $\theta$ and line segment labeled 41). 2. **Formula and Rules:** From the diagram, the line segment length $k$ can be related to the hypotenuse 41 and angle $\theta$ using trigonometric ratios. 3. **Step-by-step Solution:** 1. The line segment $k$ is the adjacent side to angle $\theta$ in the right triangle formed. 2. Using the cosine definition: $$\cos \theta = \frac{k}{41}$$ 3. Rearranging to solve for $k$: $$k = 41 \cos \theta$$ 4. **Explanation:** The cosine of an angle in a right triangle is the ratio of the adjacent side over the hypotenuse. Here, $k$ is adjacent to $\theta$, and 41 is the hypotenuse. **Final answer:** $$k = 41 \cos \theta$$