1. **Problem:** The distance between the points $(5, 2)$ and $(4, k)$ is $\sqrt{2}$. Find two possible values for $k$. 2. **Formula:** The distance $d$ between two points $(x_1, y_1)$ and $(x_2, y_2)$ is given by: $$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$$ 3. **Apply the formula:** Here, $d = \sqrt{2}$, $(x_1, y_1) = (5, 2)$, and $(x_2, y_2) = (4, k)$. $$\sqrt{2} = \sqrt{(4 - 5)^2 + (k - 2)^2}$$ 4. **Square both sides to eliminate the square root:** $$2 = (4 - 5)^2 + (k - 2)^2$$ 5. **Simplify:** $$(4 - 5)^2 = (-1)^2 = 1$$ $$2 = 1 + (k - 2)^2$$ 6. **Isolate the squared term:** $$(k - 2)^2 = 2 - 1 = 1$$ 7. **Take the square root of both sides:** $$k - 2 = \pm 1$$ 8. **Solve for $k$:** - If $k - 2 = 1$, then $k = 3$ - If $k - 2 = -1$, then $k = 1$ **Final answer:** The two possible values for $k$ are $1$ and $3$.