Question

Describe the series of transformations that have occurred to move the triangle in Quadrant IV to the triangle in Quadrant II to show that the triangles are congruent. rotation, then translation
rotation, then translation

translation, then rotation
translation, then rotation

rotation, then reflection
rotation, then reflection

reflection, then translation

Answer 1

To move a triangle from Quadrant IV to Quadrant II and show that the triangles are congruent, we can consider the following series of transformations. Let's break down a possible approach using the options you've provided:

1. **Reflection**, then **Translation**:
- **Reflection**: First, we can reflect the triangle across the y-axis. This transforms the triangle from Quadrant IV, where both x and y coordinates are negative (i.e., x < 0 and y < 0), to Quadrant III (where x is negative and y is positive).
- **Translation**: Next, we can translate the triangle in a positive x direction to move it from Quadrant III to Quadrant II, where both x and y coordinates are positive (i.e., x > 0 and y > 0).

These transformations can be mathematically expressed as follows:
- If the original triangle in Quadrant IV has vertices (x1, y1), (x2, y2), and (x3, y3), reflecting across the y-axis will change these points to (-x1, y1), (-x2, y2), and (-x3, y3).
- A subsequent translation by a certain amount (for example, adding a constant value to the x-coordinates) will shift the points into Quadrant II.

After executing this combination of a reflection followed by a translation, the resulting triangle in Quadrant II will maintain the same size and shape as the original triangle from Quadrant IV, thereby confirming that the triangles are congruent.

Thus, the final answer to show the transformations is:
**Reflection, then translation.**