Question

pentagon tuvwx is the image of pentagon tuvwx under a reflection across the line y = x followed by a translation.
write the rules for the reflection and translation.
reflection: (x, y) ↦ (\boxed{ }, \boxed{ })
translation: (x, y) ↦ (\boxed{ }, \boxed{ })

Explanation:

Step1: Determine the reflection rule

The rule for reflection across the line \(y = x\) is \((x,y)\to(y,x)\).

Step2: Find the translation rule

Let's take a point, say \(U(7,6)\). After reflection across \(y = x\), it becomes \((6,7)\). The image of \(U\) is \(U'(-2,5)\).
To get from \((6,7)\) to \((-2,5)\):
For the \(x -\)coordinate: \(6 + a=-2\), so \(a=-8\)
For the \(y -\)coordinate: \(7 + b = 5\), so \(b=-2\)
The translation rule is \((x,y)\to(x - 8,y-2)\)

Answer:

Reflection: \((x,y)\to(y,x)\)
Translation: \((x,y)\to(x - 8,y - 2)\)