The point $P(5, -2)$ is reflected about the line $y = x$ to obtain point $P'$. What are the coordinates of $P'$?

Reflection about the line $y = x$ swaps the $x$- and $y$-coordinates of any point. So $(a, b)$ becomes $(b, a)$. Applied to $P(5, -2)$: $P' = (-2, 5)$. Why: the line $y = x$ is the locus of points where $x$- and $y$-coordinates are equal. The reflection therefore *exchanges* the roles of the two coordinates. (Compare with reflection about the $x$-axis, which negates only $y$, and reflection about the $y$-axis, which negates only $x$.) - Trap A is reflection about the $x$-axis (only the sign of $y$ flips). - Trap B is reflection about the $y$-axis (only the sign of $x$ flips). - Trap D is reflection about the **origin** (both signs flip), often confused with reflection about $y = x$. Quick check: $P'$ should lie *as far on the opposite side of $y = x$* as $P$ does. Plot mentally: $P = (5, -2)$ is below the line $y = x$, and $(-2, 5)$ is the same perpendicular distance above it. ✓