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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 translation
rotation, then translation

translation, then rotation
translation, then rotation

reflection, then translation
reflection, then translation

rotation, then reflection
rotation, then reflection
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Answer 1

To determine the series of transformations that move the triangle in Quadrant IV to the triangle in Quadrant II, we need to analyze the positions of the two triangles.

1. Triangle in Quadrant IV:

   • Points: \( (1, -3) \), \( (3, -3) \), \( (1, -7) \)

2. Triangle in Quadrant II:

   • Points: \( (-5, 3) \), \( (-1, 3) \), \( (-5, 5) \)

To visualize the relationship between the two triangles:

• The Triangle in Quadrant IV lies below the x-axis, while the Triangle in Quadrant II lies above the x-axis. Thus, a reflection across the x-axis would send the triangle from Quadrant IV to a triangle positioned above the x-axis.

• Reflecting the Triangle in Quadrant IV across the x-axis gives us the following new coordinates:

  • \( (1, -3) \) becomes \( (1, 3) \)
  • \( (3, -3) \) becomes \( (3, 3) \)
  • \( (1, -7) \) becomes \( (1, 7) \)

Next, we see that the reflected triangle needs to be translated to match the positions of the Triangle in Quadrant II.

• The reflected triangle has the points \( (1, 3) \), \( (3, 3) \), \( (1, 7) \) and these can be translated to the left:
  • Translating \( (1, 3) \) to \( (-5, 3) \)
  • Translating \( (3, 3) \) to \( (-1, 3) \)
  • Translating \( (1, 7) \) to \( (-5, 5) \)

From the above analysis, we can see that the necessary transformations to get from the Triangle in Quadrant IV to the Triangle in Quadrant II are:

• Reflection across the x-axis, followed by
• Translation to the left.

Therefore, the correct answer is:

reflection, then translation.