Question

what is the correct multiple transformations that causes triangle 1 to end up at triangle 3?
\\(y = x\\)
\\(r_{y = x}\\) line and then \\(t_{(5,1)})
rotation 90 degrees cw and then translation 1 right and 5 up.
reflection over y = x line and the translation 1 right and 5 up.
reflection over y = x line and translation 1 right and 3 up.

Explanation:

Step1: Analyze reflection

Notice that triangle 1 and triangle 3 have a mirror - like relationship with respect to the line $y = x$. A reflection over the line $y=x$ swaps the $x$ and $y$ coordinates of each point of triangle 1.

Step2: Analyze translation

After reflection over $y = x$, we need to translate the reflected triangle. By comparing the positions of key points, we find that we need to move 1 unit to the right and 5 units up.

Answer:

Reflection over y = x line and then Translation 1 Right and 5 Up.