Question: On a math test Nikol needs to describe the motions which map Upper DeltaEFG to Upper DeltaMNO. The coordinates of Upper DeltaEFG are E(5,4), F(7,4), and G(5,8). The coordinates of Upper DeltaMNO are M(0,0), N(-2,0), and O(0,4). She says there is a translation 5 units to the left and 4 units down, followed by a reflection across the x-axis. Describe the sequence of rigid motions. What is Nikol's likely error? Which of the following is the description of the rigid motions which maps Upper DeltaEFG to Upper DeltaMNO? A. A translation 4 units to the left and 3 units down, followed by a reflection across the y-axis B. A translation 5 units to the left and 4 units down, followed by a reflection across the y-axis C. A translation 5 units to the left and 4 units down, followed by a reflection across the x-axis

1. **State the problem:** We want to find the sequence of rigid motions (translations and reflections) that map triangle EFG with vertices E(5,4), F(7,4), G(5,8) to triangle MNO with vertices M(0,0), N(-2,0), O(0,4). 2. **Analyze Nikol's proposed motions:** Nikol says translate 5 units left and 4 units down, then reflect across the x-axis. 3. **Apply the translation:** - Translate each point by $(-5,-4)$: - $E(5,4) \to E'(5-5,4-4) = (0,0)$ - $F(7,4) \to F'(7-5,4-4) = (2,0)$ - $G(5,8) \to G'(5-5,8-4) = (0,4)$ 4. **Apply reflection across the x-axis:** - Reflection across x-axis changes $(x,y)$ to $(x,-y)$: - $E'(0,0) \to E''(0,-0) = (0,0)$ - $F'(2,0) \to F''(2,-0) = (2,0)$ - $G'(0,4) \to G''(0,-4) = (0,-4)$ 5. **Compare with target points:** - Target points are $M(0,0)$, $N(-2,0)$, $O(0,4)$. - After motions, points are $E''(0,0)$, $F''(2,0)$, $G''(0,-4)$. - $F''$ and $G''$ do not match $N$ and $O$. 6. **Identify Nikol's error:** - The reflection should be across the y-axis, not the x-axis. 7. **Check reflection across y-axis:** - Reflection across y-axis changes $(x,y)$ to $(-x,y)$: - $E'(0,0) \to (0,0)$ - $F'(2,0) \to (-2,0)$ - $G'(0,4) \to (0,4)$ - These match $M$, $N$, and $O$ exactly. 8. **Conclusion:** - The correct sequence is translation 5 units left and 4 units down, followed by reflection across the y-axis. **Answer:** Option B is correct. **Nikol's likely error:** She reflected across the x-axis instead of the y-axis.