An illustration of a coordinate plane with four quadrants shows the x and y axes extending from negative 10 to 10 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 3 comma 1 right parenthesis, left parenthesis 3 comma 3 right parenthesis, and left parenthesis 7 comma 1 right parenthesis. The second triangle is placed in the second quadrant with vertices marked as closed points at the following coordinates: left parenthesis negative 1 comma 3 right parenthesis, left parenthesis negative 5 comma 3 right parenthesis, and left parenthesis negative 5 comma 5 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 3 right parenthesis, left parenthesis 1 comma negative 7 right parenthesis, and left parenthesis 3 comma negative 3 right parenthesis. An arrow goes from the point left parenthesis 3 comma 1 right parenthesis to left parenthesis negative 5 comma 3 right parenthesis; an arrow goes from the point left parenthesis 3 comma 3 right parenthesis to left parenthesis negative 5 comma 5 right parenthesis; an arrow goes from left parenthesis 7 comma 1 right parenthesis to left parenthesis negative 1 comma 3 right parenthesis.

Question

describe 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

reflection, then translation

translation, then reflection

translation, then transformation

rotation, then translation

Answer 1

To show that the triangles in Quadrant IV and Quadrant II are congruent, we can analyze the transformations that would align the triangle in Quadrant IV with the triangle in Quadrant II.

The triangle in Quadrant IV has vertices at the following coordinates:

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

The triangle in Quadrant II has vertices at the following coordinates:

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

To move the triangle in Quadrant IV to the position of the triangle in Quadrant II, the following sequence of transformations can be performed:

1. Reflection: First, reflect the triangle across the x-axis. This will change the y-coordinates of the vertices in Quadrant IV from negative to positive. The new coordinates after reflection will be:

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

2. Translation: Next, translate the triangle horizontally to the left. Specifically, you can move it 2 units to the left (subtracting 2 from the x-coordinates of the reflected triangle's vertices). The resulting coordinates would be:

   • \( (1-2, 3) \) → \( (-1, 3) \)
   • \( (1-2, 7) \) → \( (-1, 7) \)
   • \( (3-2, 3) \) → \( (1, 3) \)

Now, you can see that these new coordinates correspond to the vertices of the triangle in Quadrant II when considering the side lengths and angles.

Therefore, the sequence of steps to move the triangle in Quadrant IV to the triangle in Quadrant II to show that they are congruent is reflection, then translation.