An illustration of a coordinate plane with four quadrants shows the x and y axes extending from negative 5 to 5 in increments of 1. Three triangles are plotted on the graph. The first triangle is placed in the first quadrant with vertices marked as closed points at the following coordinates: left parenthesis 1 comma 1 right parenthesis, left parenthesis 3 comma 1 right parenthesis, and left parenthesis 2 comma 3 right parenthesis. The second triangle is placed in the second quadrant with vertices marked as closed points at the following coordinates: left parenthesis negative 2 comma 2 right parenthesis, left parenthesis negative 3 comma 4 right parenthesis, and left parenthesis negative 4 comma 2 right parenthesis. The third triangle is placed in the fourth quadrant with vertices marked as closed points at the following coordinates: left parenthesis 1 comma negative 1 right parenthesis, left parenthesis 1 comma negative 3 right parenthesis, and left parenthesis 3 comma negative 2 right parenthesis. An arrow goes from the point left parenthesis 1 comma 1 right parenthesis to left parenthesis negative 4 comma 2 right parenthesis; an arrow goes from the point left parenthesis 2 comma 3 right parenthesis to left parenthesis negative 3 comma 4 right parenthesis; an arrow goes from left parenthesis 3 comma 1 right parenthesis to left parenthesis negative 2 comma 2 right parenthesis.

Question

What is the sequence of steps used to move the triangle in Quadrant IV to the triangle in Quadrant II to show that the triangles are congruent?
translation, then transformation
translation, then transformation

translation, then reflection
translation, then reflection

reflection, then translation
reflection, then translation

rotation, then translation

Answer 1

To move the triangle in Quadrant IV to the triangle in Quadrant II and demonstrate that they are congruent, we need to analyze the movement of the points from the triangle in Quadrant IV to the triangle in Quadrant II.

1. The triangle in Quadrant IV has vertices at:

   • \( (1, -1) \)
   • \( (1, -3) \)
   • \( (3, -2) \)

2. The triangle in Quadrant II has vertices at:

   • \( (-2, 2) \)
   • \( (-3, 4) \)
   • \( (-4, 2) \)

To transform the triangle in Quadrant IV to the triangle in Quadrant II, we can consider the following steps:

Step 1: Reflection

First, we can reflect the triangle across the y-axis. This will change the x-coordinates of the triangle in Quadrant IV to negative values, effectively moving the triangle from Quadrant IV to Quadrant III.

• After reflecting:
  • \( (1, -1) \) becomes \( (-1, -1) \)
  • \( (1, -3) \) becomes \( (-1, -3) \)
  • \( (3, -2) \) becomes \( (-3, -2) \)

Step 2: Translation

Next, we translate the reflected triangle from Quadrant III to Quadrant II. We need to move it upwards and to the left.

• For example, translating \( (-1, -1) \) to \( (-2, 2) \) requires moving to the left by 1 unit and up by 3 units.
• This same type of movement applies to the other points as well:
  • \( (-1, -3) \) translates to \( (-3, 4) \)
  • \( (-3, -2) \) translates to \( (-4, 2) \)

Conclusion

Thus, the correct sequence of steps to move the triangle in Quadrant IV to the triangle in Quadrant II is reflection, then translation.

The correct answer is: reflection, then translation.