1. Problem: Find two possible values for $k$ given the distance between points $(5, 2)$ and $(4, k)$ is $\sqrt{2}$. 2. Formula: The distance 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. Substitute the known values: $$\sqrt{2} = \sqrt{(4 - 5)^2 + (k - 2)^2}$$ 4. Square both sides to eliminate the square root: $$2 = (4 - 5)^2 + (k - 2)^2$$ 5. Simplify the right side: $$(4 - 5)^2 = (-1)^2 = 1$$ 6. So, $$2 = 1 + (k - 2)^2$$ 7. Subtract 1 from both sides: $$2 - 1 = (k - 2)^2$$ $$1 = (k - 2)^2$$ 8. Take the square root of both sides: $$\sqrt{1} = \pm (k - 2)$$ $$1 = \pm (k - 2)$$ 9. Solve for $k$: - Case 1: $$1 = k - 2 \implies k = 3$$ - Case 2: $$-1 = k - 2 \implies k = 1$$ 10. Final answer: The two possible values for $k$ are $1$ and $3$.