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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

reflection, then translation

translation, then rotation

rotation, then translation

rotation, then reflection

1 answer

To determine the series of transformations that have occurred to move the triangle in Quadrant IV to the triangle in Quadrant II, we need to analyze their positions and how they relate to one another.

  1. Identifying the triangles:

    • Triangle in Quadrant IV:
      • Points: (1, -3), (3, -3), (1, -7)
    • Triangle in Quadrant II:
      • Points: (-5, 3), (-1, 3), (-5, 5)

  2. Finding the reflection:

    • The triangle in Quadrant IV can be reflected over the y-axis to move it closer to Quadrant II. Reflecting the points:
      • (1, -3) reflects to (-1, -3)
      • (3, -3) reflects to (-3, -3)
      • (1, -7) reflects to (-1, -7)

    However, these points do not match the corresponding points of the triangle in Quadrant II directly after reflection.

  3. Translation:

    • After reflection, we need to translate the reflected triangle upward. The reflected points can be further moved up by a vertical translation.
    • Translating the points:
      • (-1, -3) moved to (-1, 3)
      • (-3, -3) moved to (-5, 3)
      • (-1, -7) moved to (-5, 5)

  4. Conclusion:

    • From Quadrant IV to Quadrant II, we first reflect the triangle over the y-axis (moving it from Quadrants IV to III) and then translate it upward to position it in Quadrant II.

Therefore, the correct series of transformations is:

reflection, then translation.