Question

Write a rule for the glide reflection that maps △ABC with vertices A(−4, −2), B(−2, 6), and C(4, 4) to △A′B′C′ with vertices A′(4, 2), B′(6, −6), and C′(12, −4).

Answer 1

The rule for the glide reflection mapping △ABC to △A′B′C′ is a translation by (8, 4) followed by a reflection across the y-axis.

Step-by-step explanation:

To find the rule for the glide reflection that maps △ABC to △A′B′C′, we need to identify the translation and reflection that would map one triangle onto the other. A glide reflection is a composite transformation consisting of a translation followed by a reflection across a line that is parallel to the direction of the translation.

Firstly, we determine the translation vector by finding the difference between the coordinates of corresponding points, for example, A and A′. The translation vector would be (8, 4).

Next, to find the line of reflection, we observe that the midpoints between corresponding points should lie on this line. For instance, the midpoint M between A and A′ is at ((-4 + 4) / 2, (-2 + 2) / 2) = (0,0). This is the origin, and we see that B′ and C′ are also reflections of B and C across the y-axis. Thus, the line of reflection is the y-axis.

The glide reflection that maps △ABC to △A′B′C′ can be described as a translation by (8, 4) followed by a reflection across the y-axis.