The series of transformations correctly maps △ABC to △XYZ the correct series of transformations that maps △ABC to △XYZ is to dilate △ABC by a scale factor of 3 centered at the origin, then translate the result down 5 units.
To determine the correct series of transformations that maps △ABC to △XYZ, we need to compare the corresponding sides and angles of the two triangles. If we can show that the two triangles are similar, then we can use a dilation to map one triangle onto the other.
Let's assume that △ABC is mapped to △XYZ by a dilation centered at the origin with scale factor k, followed by a translation down 5 units. This can be written as:
△ABC → Dilate(k) → △A'B'C' → Translate down 5 units → △XYZ
To show that △ABC and △XYZ are similar, we need to show that their corresponding angles are congruent, and their corresponding sides are proportional.
Let's first consider the angles. We are not given any information about the angles of the two triangles, so we assume that they are the same. Therefore, we only need to show that their corresponding sides are proportional.
The dilation centered at the origin scales all distances by a factor of k. Therefore, the length of each side of △A'B'C' is k times the length of the corresponding side of △ABC. Then, the translation down 5 units shifts all points 5 units downwards. Therefore, the length of each side of △XYZ is equal to the length of the corresponding side of △A'B'C' minus 5 units.
From this analysis, we can conclude that the sides of △ABC and △XYZ are proportional, with a scale factor of k, and therefore, △ABC and △XYZ are similar triangles.
Now, let's look at the answer choices and see which one fits the description of a dilation centered at the origin with a scale factor of k, followed by a translation down 5 units:
Dilate △ABC by a scale factor of 3 centered at the origin, then translate the result down 5 units.
Dilate △ABC by a scale factor of 13 centered at the origin, then translate the result down 5 units.
Dilate △ABC by a scale factor of 1/3 centered at the origin, then translate the result down 5 units.
Dilate △ABC by a scale factor of 13 centered at the origin, then reflect the result across the x-axis.
Dilate △ABC by a scale factor of 1/3 centered at the origin, then reflect the result across the x-axis.
Dilate △ABC by a scale factor of 13 centered at the origin, then rotate the result 90 degrees clockwise about the origin.
Dilate △ABC by a scale factor of 1/3 centered at the origin, then rotate the result 90 degrees clockwise about the origin.
The only answer choice that fits the description of a dilation centered at the origin with a scale factor of k, followed by a translation down 5 units, is the first one:
Dilate △ABC by a scale factor of 3 centered at the origin, then translate the result down 5 units.
Therefore, the correct series of transformations that maps △ABC to △XYZ is to dilate △ABC by a scale factor of 3 centered at the origin, then translate the result down 5 units.