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Question

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.
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)
Responses

rotation, then reflection
rotation, then reflection

reflection, then translation
reflection, then translation

rotation, then translation
rotation, then translation

translation, then rotation

Answer 1

To analyze the transformation from the triangle in Quadrant IV to the triangle in Quadrant II, we need to consider the positions of the corresponding points:

• Quadrant IV triangle vertices: \( (1, -3) \), \( (3, -3) \), \( (1, -7) \)
• Quadrant II triangle vertices: \( (-5, 3) \), \( (-1, 3) \), \( (-5, 5) \)

A good approach to understand the congruence and transformation is to check for geometric relationships, particularly if reflections or rotations can map one set of points to the other.

1. Reflection: The triangles could be reflected across the x-axis, which would change the y-coordinates of the Quadrant IV points to their opposites.

   • Reflecting \( (1, -3) \) across the x-axis gives \( (1, 3) \).
   • Reflecting \( (3, -3) \) gives \( (3, 3) \).
   • Reflecting \( (1, -7) \) gives \( (1, 7) \).

   However, this does not map directly to the Quadrant II triangle.

2. Translation: After finding the possible reflections, we need to check if a translation can align the reflected points with the target Quadrant II triangle.

   To align, we would need to translate:

   • If we reflect then shift down (essentially a transformation of more than one step), this may be too complex.

3. Verifying via Rotation: Let's consider a 180-degree rotation about the origin followed by a translation.

   • A rotation of 180 degrees would transform Quadrant IV to Quadrant II:

     • For \( (1, -3) \) → \( (-1, 3) \)
     • For \( (3, -3) \) → \( (-3, 3) \)
     • For \( (1, -7) \) → \( (-1, 7) \)

   • Now we could translate those points, which would be the final transformation to get to the positions of the second triangle.

Ultimately, it seems the key transformation that achieves this result with the fewest steps is reflection, then translation. The most fitting response is:

reflection, then translation