Question

graph (\triangle mpq) with vertices (m(-4,3)), (p(-5,8)), and (q(-1,6)) and its image after the glide reflection with a translation along (langle - 4,-4
angle) and a reflection in the line (y = x).

Explanation:

Step1: Translation

For a point \((x,y)\) translated along \(\langle a,b
angle\), the new point is \((x + a,y + b)\).
For \(M(-4,3)\): \(x=-4,a = - 4,y = 3,b=-4\), new point \(M'=(-4-4,3 - 4)=(-8,-1)\)
For \(P(-5,8)\): \(x=-5,a=-4,y = 8,b=-4\), new point \(P'=(-5-4,8 - 4)=(-9,4)\)
For \(Q(-1,6)\): \(x=-1,a=-4,y = 6,b=-4\), new point \(Q'=(-1-4,6 - 4)=(-5,2)\)

Step2: Reflection in the line \(y=x\)

For a point \((x,y)\) reflected in the line \(y = x\), the new point is \((y,x)\)
For \(M'(-8,-1)\): new point \(M''=(-1,-8)\)
For \(P'(-9,4)\): new point \(P''=(4,-9)\)
For \(Q'(-5,2)\): new point \(Q''=(2,-5)\)

Answer:

The vertices of the original triangle \(\triangle MPQ\) are \(M(-4,3)\), \(P(-5,8)\), \(Q(-1,6)\). After translation along \(\langle-4,-4
angle\), the vertices are \(M'(-8,-1)\), \(P'(-9,4)\), \(Q'(-5,2)\). After reflection in the line \(y = x\), the vertices of the image triangle are \(M''(-1,-8)\), \(P''(4,-9)\), \(Q''(2,-5)\)