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
reflection, then translation

Answer 1

To determine the congruence between the triangles in Quadrant IV and Quadrant II and the sequence of transformations that can be used to move one triangle onto the other, we analyze the coordinates of their vertices:

• Quadrant IV triangle vertices: (1, -1), (1, -3), (3, -2)
• Quadrant II triangle vertices: (-2, 2), (-3, 4), (-4, 2)

To move the triangle in Quadrant IV to the triangle in Quadrant II, the steps needed are as follows:

1. Reflection: Reflect the triangle across the y-axis. This changes the x-coordinates from positive to negative. For example:

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

2. Translation: After reflection, translate the triangle to align it with the triangle in Quadrant II:

   • Move the reflected vertices from their current positions to the target positions. For example, translating by moving left by 1 and up by 2 will align the vertices with those of the triangle in Quadrant II.

Hence, the sequence of steps used to move the triangle in Quadrant IV to the triangle in Quadrant II is:

reflection, then translation.