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.(1 point)

A coordinate plane with 4 quadrants shows x and y axes ranging from negative 8 to 8 in increments of 1. Three triangles are formed by joining three plotted points each. Two of the triangles are joined by connecting their vertices with rays. The coordinates of the plotted points for the triangle in quadrant 4 are left parenthesis 1 comma negative 3 right parenthesis, left parenthesis 3 comma negative 3 right parenthesis, and left parenthesis 1 comma negative 7 right parenthesis. The coordinates for the triangle in quadrant 2 are as follows left parenthesis negative 5 comma 3 right parenthesis, left parenthesis negative 1 comma 3 right parenthesis, and left parenthesis negative 5 comma 5 right parenthesis. The side joining the vertices at left parenthesis negative 5 comma 3 right parenthesis and left parenthesis negative 5 comma 5 right parenthesis is labeled as e. The side joining the vertices at left parenthesis negative 5 comma 5 right parenthesis and left parenthesis negative 1 comma 3 right parenthesis is labeled as g. The side joining the vertices at left parenthesis negative 5 comma 3 right parenthesis and left parenthesis negative 1 comma 3 right parenthesis is labeled as h. The coordinates of the plotted points for the triangle in quadrant 1 are left parenthesis 3 comma 1 right parenthesis, left parenthesis 7 comma 1 right parenthesis, and left parenthesis 3 comma 3 right parenthesis. Three rays join the complementary vertices of the triangles in quadrants 1 and 2. A ray connects point left parenthesis 3 comma 1 right parenthesis and the point left parenthesis negative 5 comma 3 right parenthesis. A ray connects point left parenthesis 7 comma 1 right parenthesis and the point left parenthesis negative 1 comma 3 right parenthesis. A ray connects point left parenthesis 3 comma 3 right parenthesis and the point left parenthesis negative 5 comma 5 right parenthesis. The arrow heads are on the vertices of the triangle in quadrant 2.

translation, then rotation
translation, then rotation

reflection, then translation
reflection, then translation

rotation, then translation
rotation, then translation

rotation, then reflection
rotation, then reflection

Answer 1

To prove that the triangle in Quadrant IV is congruent to the triangle in Quadrant II, we need to describe the series of transformations that can map one triangle onto the other.

1. **Reflection**: The first transformation is a **reflection** over the y-axis.
- The triangle in Quadrant IV has vertices at (1, -3), (3, -3), and (1, -7). Reflecting these points over the y-axis will result in:
- (1, -3) → (-1, -3)
- (3, -3) → (-3, -3)
- (1, -7) → (-1, -7)

2. **Translation**: The second transformation is a **translation**.
- We can move the reflected triangle from its current position in Quadrant III to the coordinates of the triangle in Quadrant II.
- To transform (-3, -3) to (-1, 3), you can translate by adding 2 in the x-direction and 6 in the y-direction.
- Similarly, for the other points:
- (-1, -3) translates to (-5, 3) (moving left by 4 and up by 6).
- (-1, -7) translates to (-5, 5) (moving left by 4 and up by 12).

By performing these two transformations – reflecting over the y-axis and then translating – we can map the triangle in Quadrant IV to the triangle in Quadrant II.

Thus, the answer is:
**reflection, then translation**.