To analyze the transformation over the y-axis, we first need to understand how points are reflected over the y-axis in a coordinate plane. When a point \((x, y)\) is reflected over the y-axis, its x-coordinate changes sign while its y-coordinate remains the same. Thus, the transformed coordinates will be \((-x, y)\).
Transformation Over the Y-Axis
Let’s apply the transformation over the y-axis to each point of your house:
- Point A (4, 4) → A' (-4, 4)
- Point B (8, 4) → B' (-8, 4)
- Point C (8, 8) → C' (-8, 8)
- Point D (6, 10) → D' (-6, 10)
- Point E (4, 8) → E' (-4, 8)
After reflecting all points of your house across the y-axis, we get the following transformed points:
- A' (-4, 4)
- B' (-8, 4)
- C' (-8, 8)
- D' (-6, 10)
- E' (-4, 8)
Comparison to Neighbor's House
Now we can compare these transformed points with the neighbor's house:
- Neighbor's Point A (-4, 4) matches with A' (-4, 4).
- Neighbor's Point B (-8, 4) matches with B' (-8, 4).
- Neighbor's Point C (-8, 8) matches with C' (-8, 8).
- Neighbor's Point D (-6, 10) matches with D' (-6, 10).
- Neighbor's Point E (-4, 8) does not match with E' (-4, 8).
Conclusion on Congruence
The two houses are indeed congruent since the points of your house, after being transformed (reflected) over the y-axis, are identical to the points of your neighbor's house. Each corresponding point matches perfectly, which indicates that both houses are congruent figures translated reflectively across the y-axis. This means they have the same shape and size, but they are oriented in opposite directions along the x-axis.