To determine the series of transformations to move a triangle in Quadrant IV to a triangle in Quadrant II and show that the triangles are congruent, we need to analyze the positions of the triangles in the Cartesian coordinate system.
- Quadrant IV: In this quadrant, both x-coordinates are positive and y-coordinates are negative.
- Quadrant II: In this quadrant, x-coordinates are negative and y-coordinates are positive.
Given that congruent triangles can be moved from one position to another through a series of transformations such as translations, rotations, reflections, or a combination thereof, we can consider the necessary changes in the coordinates:
- To move from Quadrant IV to Quadrant II, the triangle must change positions such that:
- The x-coordinates change from positive to negative, suggesting a reflection across the y-axis.
- The y-coordinates change from negative to positive, indicating that points must be moved upward.
The most efficient way to achieve this is:
- Reflection across the y-axis: This will turn the triangle in Quadrant IV into a triangle that is still in Quadrant III (thus still below the x-axis but also now on the negative side of the x-axis).
- Translation upward: This will move the triangle from Quadrant III to Quadrant II, making its y-coordinates positive.
Thus, the series of transformations that occurred to move the triangle in Quadrant IV to the triangle in Quadrant II is:
Reflection, then translation.