Question

trapezoid klmn is the image of trapezoid klmn under a translation followed by a reflection across the line y = x.
write the rules for the translation and reflection.
translation: (x, y) ↦ (□, □)
reflection: (x, y) ↦ (□, □)

Explanation:

Step1: Find the translation rule

Let's consider a point, say \(K(2,-2)\) and \(K'(1,6)\).
First, assume the translation rule is \((x,y)\to(x + a,y + b)\).
For \(x\)-coordinate: \(2+a = 1\Rightarrow a=-1\).
For \(y\)-coordinate: \(-2 + b=6\Rightarrow b = 8\).
So the translation rule is \((x,y)\to(x-1,y + 8)\).

Step2: Find the reflection rule

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

Answer:

Translation: \((x,y)\to(x - 1,y+8)\)
Reflection: \((x,y)\to(y,x)\)