Question

which statements, when combined, describe a sequence of transformations that exhibits the similarity between δqrs and δqrs? a a translation: (x, y) ↦ (x − 5, y + 8). b a translation: (x, y) ↦ (x + 5, y + 8). c a translation: (x, y) ↦ (x + 8, y + 5). d a dilation centered at the origin by a scale factor of \\(\frac{1}{3}\\). e a dilation centered at the origin by a scale factor of 3.

Explanation:

1. First, analyze the dilation: The triangle \( \triangle QRS \) (orange) is smaller than \( \triangle Q'R'S' \) (blue), so the dilation scale factor should be greater than 1. A dilation centered at the origin with scale factor 3 (option E) would enlarge the smaller triangle to match the size of the larger one.
2. Then, analyze the translation: To map the dilated (or original) smaller triangle to the larger one, we check the horizontal and vertical shifts. From the coordinates, moving from the smaller triangle's position to the larger one, we move 5 units right (\( x + 5 \)) and 8 units up (\( y + 8 \)), which is the translation in option B.

Answer:

B. A translation: \( (x, y) \mapsto (x + 5, y + 8) \)
E. A dilation centered at the origin by a scale factor of 3.