Questions: Error Analysis Hugo graphed triangle PQR and (Rt · T(3,1))(triangle PQR) where the equation of line t is y=2. His translation and reflection were both correct. What mistake did Hugo make? (G) MP. 3

Solution Steps

The notation $R_t \circ T_{(3,1)}(\triangle PQR)$ represents a sequence of transformations applied to triangle $\triangle PQR$. First, the triangle is translated by the vector $\langle 3,1 \rangle$, denoted by $T_{(3,1)}$. Then, the translated triangle is reflected across the line $t$ with equation $y=2$, denoted by $R_t$.

The original triangle $\triangle PQR$ has vertices approximately at $P(-3,4)$, $Q(-2,4)$, and $R(-2,2)$. The problem states both the translation and reflection were done correctly. $\triangle P'Q'R'$ has vertices approximately at $P'(-2,1)$, $Q'(1,1)$, and $R'(1,3)$.

The vertices of the translated triangle, before reflection, should be $P''(0,5)$, $Q''(1,5)$, and $R''(1,3)$. When reflected across $y=2$, the y-coordinates change according to the formula $y'= 2-(y-2) = 4-y$. So, the reflected vertices should be $P'''(0,-1)$, $Q'''(1,-1)$, and $R'''(1,1)$. The graph incorrectly places $P'$ at $(-2,1)$ instead of $(0,-1)$. $Q'$ is placed at $(1,-1)$ rather than at $(1,1)$. $R'$ is placed at $(1,3)$.

Hugo performed the transformations in the wrong order. He reflected the triangle $\triangle PQR$ over $y=2$, then translated it by $\langle 3, 1 \rangle$, while the prompt asks to first translate then reflect the triangle.

Final Answer: