Okay, let's break this down step-by-step:
1. To make Triangle ABC congruent to Triangle FGH, we need to translate point F so that side lengths AB, BC, and CA map to FG, GH, and HF respectively.
2. From the given side length formulas, we can see that:
AB = BC = CA = sqrt(2)
FG = sqrt(5)
GH = sqrt(2)
3. To map AB to FG, and BC to GH, point F should be plotted at (1,2). This will make FG = sqrt(5) and GH = sqrt(2), matching AB and BC respectively.
4. With point F at (1,2), HF will also equal sqrt(2), matching CA. So Triangle ABC will be congruent to Triangle FGH.
5. The sequence of rigid motions:
- Translate Triangle ABC so that point A maps to point F (1,2)
- Reflect the translated triangle over line FG
6. By the SSS (Side-Side-Side) Congruence Theorem, Triangle ABC is congruent to Triangle FGH, since all corresponding sides (AB & FG, BC & GH, CA & HF) are equal.
Therefore, plotting point F at (1,2) and applying the translation and reflection described will map Triangle ABC onto Triangle FGH, making the triangles congruent by the SSS theorem.