Question

given the line segment cd with coordinates c(2, -5) and d(4, 0), find the coordinates of the final image after the following glide reflection:
translation: $(x,y)\to(x + 2,y + 2)$
reflection: $y=x$
c(
d(

Explanation:

Step1: Apply translation to point C

For point \(C(2,-5)\), using the translation \((x,y)\to(x + 2,y+2)\).
The \(x\)-coordinate becomes \(2+2 = 4\), and the \(y\)-coordinate becomes \(-5 + 2=-3\). So the translated point \(C'\) (before reflection) is \((4,-3)\).

Step2: Reflect translated point C over \(y = x\)

For a reflection over \(y=x\), the rule is \((x,y)\to(y,x)\).
For the point \((4,-3)\), after reflection, the \(x\) and \(y\) - coordinates swap. So \(C''\) is \((-3,4)\).

Step3: Apply translation to point D

For point \(D(4,0)\), using the translation \((x,y)\to(x + 2,y+2)\).
The \(x\)-coordinate becomes \(4+2=6\), and the \(y\)-coordinate becomes \(0 + 2=2\). So the translated point \(D'\) (before reflection) is \((6,2)\).

Step4: Reflect translated point D over \(y = x\)

For a reflection over \(y=x\), the rule is \((x,y)\to(y,x)\).
For the point \((6,2)\), after reflection, the \(x\) and \(y\) - coordinates swap. So \(D''\) is \((2,6)\).

Answer:

\(C''(-3,4)\), \(D''(2,6)\)