Question: Part 2 of 2
On a math test Nikoli needs to describe the motions which map $\triangle EFG$ to $\triangle MNO$. The coordinates of $\triangle EFG$ are $E(5,4)$, $F(7,4)$, and $G(5,8)$. The coordinates of $\triangle MNO$ 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 Nikoli's likely error?
Which of the following is the description of the rigid motions which maps $\triangle EFG$ to $\triangle MNO$?
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
What is Nikoli's likely error?
A. She reflected the triangle across the wrong axis.
B. She did not move the correct units down.
C. She did not move the correct units to the left.
1. **State the problem:** We need to find the sequence of rigid motions (translation and reflection) that maps $\triangle EFG$ 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 the translation:** Nikoli says the translation is 5 units left and 4 units down. Let's apply this translation to each vertex of $\triangle EFG$:
$$E'(5-5,4-4) = (0,0)$$
$$F'(7-5,4-4) = (2,0)$$
$$G'(5-5,8-4) = (0,4)$$
3. **Compare translated points to $\triangle MNO$:**
Nikoli's translated points are $E'(0,0)$, $F'(2,0)$, $G'(0,4)$.
$\triangle MNO$ points are $M(0,0)$, $N(-2,0)$, $O(0,4)$.
Notice that $F'(2,0)$ does not match $N(-2,0)$, but is its reflection across the y-axis.
4. **Reflection step:** Nikoli says the reflection is across the x-axis, but reflecting $F'(2,0)$ across the x-axis would give $(2,0)$ (no change), and $G'(0,4)$ would become $(0,-4)$, which does not match $O(0,4)$.
5. **Correct reflection axis:** Reflecting across the y-axis changes $(x,y)$ to $(-x,y)$.
Reflecting $F'(2,0)$ across the y-axis gives $(-2,0)$, matching $N(-2,0)$.
Reflecting $G'(0,4)$ across the y-axis gives $(0,4)$, matching $O(0,4)$.
6. **Conclusion:** The correct sequence is a translation 5 units left and 4 units down, followed by a reflection across the y-axis.
7. **Nikoli's likely error:** She reflected across the wrong axis (x-axis instead of y-axis).
**Answer to multiple choice:**
- The correct description is option B: "A translation 5 units to the left and 4 units down, followed by a reflection across the y-axis."
- Nikoli's likely error is option A: "She reflected the triangle across the wrong axis."