Question

triangle ( def ) is the image of triangle ( def ) under a reflection across the line ( y = x ) followed by a translation.
write the rules for the reflection and translation.
reflection: ( (x,y)\to(square,square) )
translation: ( (x,y)\to(square,square) )

Explanation:

Step1: Find 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. For example, after reflection of \(D(-8,6)\) across \(y = x\) we get \((6,-8)\). The final image \(D'(-1,-7)\).
To get from \((6,-8)\) to \((-1,-7)\), we subtract \(7\) from \(x\) - coordinate (\(6-7=-1\)) and add \(1\) to \(y\) - coordinate (\(-8 + 1=-7\)). So the translation rule is \((x,y)\to(x - 7,y+1)\)

Answer:

Reflection: \((x,y)\to(y,x)\)
Translation: \((x,y)\to(x - 7,y + 1)\)