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 occurred to move the triangle in Quadrant IV to the triangle in Quadrant II to show that the triangles are congruent.
translation, then rotation
translation, then rotation

rotation, then translation
rotation, then translation

rotation, then reflection
rotation, then reflection

reflection, then translation

Answer 1

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

1. Triangle in Quadrant IV (Coordinates):

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

2. Triangle in Quadrant II (Coordinates):

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

Analyzing Transformations

• Reflection: Since both triangles are in opposite quadrants, we need a reflection across the y-axis. Reflecting the triangle in Quadrant IV (coordinates will change signs for x-values only):

  • A(1, -3) → A′(-1, -3)
  • B(3, -3) → B′(-3, -3)
  • C(1, -7) → C′(-1, -7)

• Translation: After the reflection across the y-axis, we need to move the reflected triangle into the position of the triangle in Quadrant II. We can see the points after reflection need to be moved:

  • To get from A′(-1, -3) to D(-5, 3), we can translate left by 4 units and up by 6 units.
  • A′ moves (-1 - 4, -3 + 6) = D
  • Similarly, for points B' and C', we can check:
    • B′(-3, -3) translates to E(-1, 3)
    • C′(-1, -7) translates to F(-5, 5)

Both triangles can be shown to be congruent through these transformations.

Conclusion

The transformation to move the triangle in Quadrant IV to the triangle in Quadrant II is: reflection, then translation.

Therefore, the correct choice from your options is: reflection, then translation.