To determine which series of transformations Lyle most likely used to convert triangle ABC into triangle XYZ while keeping the triangles congruent, we need to analyze the nature of congruence and the transformations involved:
- Congruent triangles must have the same shape and size. This means that transformations that produce congruent triangles should not alter these properties.
- Types of transformations that keep triangles congruent include translations, rotations, and reflections. Dilation, however, changes the size of a figure, so it cannot be used if we want to maintain congruence.
Now, let's evaluate the options given:
A. a translation left and then a dilation with a scale factor of 3
- This transformation would increase the size of the triangle (dilation with a scale factor of 3), which would make triangle XYZ larger than triangle ABC, thus they wouldn’t be congruent.
B. a dilation with a scale factor of 2.5 and then a rotation 180˚ counterclockwise
- Similar to option A, the initial dilation would change the size, so this transformation would also not result in congruent triangles.
C. a rotation 90˚ clockwise and then a reflection across the y-axis
- Both rotation and reflection are rigid motions that maintain congruence. This transformation should maintain the size and shape of the triangle.
D. a reflection across the x-axis and then a dilation with a scale factor of 0.25
- The reflection would keep the triangle congruent, but the dilation would reduce the size, which would not allow XYZ to remain congruent to ABC.
Based on the analysis, the only valid option that will result in congruent triangles is C: a rotation 90˚ clockwise and then a reflection across the y-axis.