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
reflection, then translation

translation, then transformation
translation, then transformation

rotation, then translation
rotation, then translation

translation, then reflection

Answer 1

To show that the triangles are congruent through a sequence of transformations, we can analyze the movements of the triangles.

1. Reflection: First, reflect the triangle in Quadrant IV across the y-axis. This will map each vertex of the triangle in Quadrant IV to a corresponding vertex in Quadrant II. For example:

   • The point (1, -3) reflects to (-1, -3).
   • The point (1, -7) reflects to (-1, -7).
   • The point (3, -3) reflects to (-3, -3).

   After this reflection, the new vertices are now positioned closer to the ones in Quadrant II, but along the line y = -3.

2. Translation: Next, we need to translate the reflected triangle so that its vertices match up with the vertices of the triangle in Quadrant II. For example:

   • The point (-1, -3) is translated upward to (-1, 3).
   • The point (-1, -7) is translated upward to (-1, 5).
   • The point (-3, -3) is translated to the left to align with the vertex (-5, 3).

Through these steps, the previously placed triangle in Quadrant IV will match the one in Quadrant II, showing they are congruent.

Thus, the correct answer is:

Reflection, then translation.