Use the image to answer the question.

Question

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.

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?

(1 point)
Responses

rotation, then translation
rotation, then translation

translation, then transformation
translation, then transformation

translation, then reflection
translation, then reflection

reflection, then translation

Answer 1

To determine the sequence of steps used to move the triangle in Quadrant IV to the triangle in Quadrant II, we need to analyze the transformations required to map one triangle onto the other while maintaining congruence.

1. Reflection: The triangle in Quadrant IV, which has vertices at (1, -1), (1, -3), and (3, -2), needs to undergo a reflection over the y-axis to move to the second quadrant. The reflection over the y-axis will change the x-coordinates from positive to negative, making the coordinates of the reflected triangle: (-1, -1), (-1, -3), and (-3, -2).

2. Translation: After reflection, the triangle’s vertices will need to be translated to coincide with the coordinates of the triangle in Quadrant II at (-2, 2), (-3, 4), and (-4, 2). The translation can be applied by moving the points up and left to reach their respective locations.

Based on this analysis, the correct sequence of transformations is:

reflection, then translation.

So the correct response is:

reflection, then translation.