Problem Statement
Point $P(2, -4)$ is reflected in the $x$-axis to point $P'$. Point $P'$ is then reflected in the origin to point $P''$.
(i) Write down the coordinates of $P'$ and $P''$.
(ii) Find the length of the segment $PP''$.
(iii) Write down the single transformation that maps $P$ directly to $P''$.
Verified Solution & Marking Scheme
Find Coordinates After Reflections
1. Reflection in $x$-axis: $(x, y) \to (x, -y) \implies P'(2, 4)$.
2. Reflection in origin: $(x, y) \to (-x, -y) \implies P''(-2, -4)$.
Calculate Length PP''
$P(2, -4) \text{ and } P''(-2, -4)$
Notice the $y$-coordinates are equal ($-4$).
$\text{Length } PP'' = |2 - (-2)| = |2 + 2| = 4 \text{ units}$
Identify Single Transformation
From $P(2, -4)$ to $P''(-2, -4)$, the $x$-coordinate is negated while $y$-coordinate is unchanged: $(x, y) \to (-x, y)$. This is a reflection in the $y$-axis.