Final answer:
To determine which equations are true when triangle ABC is taken to triangle A1B1C1 through rigid motions and a dilation, we need to know if triangle ABC is isosceles and which sides are equal. Without more information, we cannot accurately select the true equation(s).
Step-by-step explanation:
The question posed by the student involves the properties of similar triangles and the effect of rigid motions and dilations on these triangles. The given triangles, ABC and A1B1C1, can be taken to each other using a combination of rotations, reflections, translations (rigid motions), and a dilation, which preserves angles and proportionality but not actual sizes. Therefore, corresponding sides of similar triangles are proportional.
From the options provided, the correct equations that represent these properties are as follows:
A. A1B1 = B1C1 / AB = BC: This equation may be true if triangle ABC is isosceles with AB = BC.
B. B1C1 = A1C1 / BC = AC: This equation would be correct if triangle ABC is isosceles with AC = BC.
C. A1C1 = B1C1 / AC = AB: This would be accurate if instead triangle ABC is isosceles with AC = AB.
D. A1B1 = A1C1 / AB = AC: This equation would hold true for an isosceles triangle ABC with AB = AC.
Without additional information about the specific nature of triangle ABC, it is not possible to definitively select the correct equation(s). Therefore, more details or clarifications about the triangles' side lengths are required to determine which equation(s) apply.