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.

Question

describe the series of transformations that have occured to move the triangle in Quadrant IV to the triangle in quadrant II to show that the triangles are congruent.

translation, then rotation
rotation, then reflection
reflection, then translation
rotation, then translation

Answer 1

To determine the sequence of transformations that will move the triangle in Quadrant IV to the triangle in Quadrant II, let's first analyze the coordinates of the vertices of both triangles.

Triangle in Quadrant IV:

• Vertex A: (1, -3)
• Vertex B: (3, -3)
• Vertex C: (1, -7)

Triangle in Quadrant II:

• Vertex D: (-5, 3)
• Vertex E: (-1, 3)
• Vertex F: (-5, 5)

Steps to Analyze Transformations:

1. Reflection:

   • Reflect the Quadrant IV triangle across the x-axis. This changes the y-coordinates of each point:
     • A (1, -3) becomes A' (1, 3)
     • B (3, -3) becomes B' (3, 3)
     • C (1, -7) becomes C' (1, 7)
   • After reflection, we don't have the correct coordinates yet, as they don't match any vertices in Quadrant II.

2. Translation:

   • After reflecting, we now translate the triangle to the left:
     • Translate A' (1, 3) to D (-5, 3): this requires moving left by 6 units.
     • Translate B' (3, 3) to E (-1, 3): this requires moving left by 4 units.
     • Translate C' (1, 7) to F (-5, 5): this is also not a matching condition.

After identifying these transformations, it appears that we may also need a rotation.

Finding the Correct Sequence of Transformations:

Let's try another sequence:

• Reflection across the y-axis (transforming (x, y) to (-x, y)):

  • A (1, -3) becomes (-1, -3)
  • B (3, -3) becomes (-3, -3)
  • C (1, -7) becomes (-1, -7)

• Then we need a translation to move to the desired vertices in Quadrant II:

  • From (-1, -3) to (-5, 3) involves a left and upward movement.
  • The correct translation needed to align with the new coordinates of the triangle in Quadrant II.

Conclusion:

The correct transformation to move the Triangle in Quadrant IV to the Triangle in Quadrant II requires reflecting across the y-axis and translating afterward.

Thus, the answer is:

Reflection, then translation.