Question

1. given a triangle, \\( \triangle abc \\) with the following vertices \\( (0,3),(4,1) \\) and \\( (3,5) \\). find new image \\( r_{y=-1} \circ t_{2,-1} \\) (4 pts)

Explanation:

Step1: Translation by \(T_{2,-1}\)

For a point \((x,y)\), the translation \(T_{2,-1}\) gives \((x + 2,y-1)\).
For vertex \((0,3)\): \((0+2,3 - 1)=(2,2)\)
For vertex \((4,1)\): \((4+2,1 - 1)=(6,0)\)
For vertex \((3,5)\): \((3+2,5 - 1)=(5,4)\)

Step2: Reflection about \(y=-1\)

The formula for reflection about the line \(y = k\) of a point \((x,y)\) is \((x,2k - y)\). Here \(k=-1\), so the formula is \((x,-2 - y)\)
For point \((2,2)\): \((2,-2-2)=(2,-4)\)
For point \((6,0)\): \((6,-2 - 0)=(6,-2)\)
For point \((5,4)\): \((5,-2-4)=(5,-6)\)

Answer:

The new vertices of the triangle are \((2,-4)\), \((6,-2)\) and \((5,-6)\)