The transformation \( T_{(3,8)} * r_{\left(90^{\circ}, A\right)} \) maps \( \triangle A B C \) to \( \triangle D E F \). Triangle \( A B C \) is ? to \( \triangle D E F \) because \( T_{(3,8)} \cdot r_{\left(90^{\circ}, A\right)} \) is a ?

The transformation \( T_{(3,8)} * r_{\left(90^{\circ}, A\right)} \) involves a translation of the triangle \( ABC \) by moving it 3 units right and 8 units up, followed by a rotation of 90 degrees around point A. This combination of transformations ensures that the shape and size of triangle \( ABC \) are preserved throughout the process. Thus, triangle \( ABC \) is congruent to triangle \( DEF \) because \( T_{(3,8)} \cdot r_{\left(90^{\circ}, A\right)} \) is a composite transformation involving translation and rotation, both of which preserve congruency. If you're intrigued by transformations further, dive into how different combinations of transformations can lead to different results and how they can be represented using matrices! Transformations aren’t just theoretical – they’re used in computer graphics, robotics, and even architecture to ensure designs are spot on. How cool is that?