Question

1. the coordinates of \\( \triangle a b c \\) are \\( a ( 2,3 ) \\), \\( b ( 6,6 ) \\), and \\( c ( 7,2 ) \\). after a translation, the image of vertex \\( a \\) is \\( a ^ { prime } ( - 6,1 ) \\).

a) give the coordinates of \\( b ^ { prime } \\) and \\( c ^ { prime } \\) after the same translation.
b) graph \\( \triangle a b c \\) and \\( \triangle a ^ { prime } b ^ { prime } c ^ { prime } \\) on the graph above and to the right.

Explanation:

Step1: Determine the translation rule

The translation from \(A(2,3)\) to \(A'(-6,1)\) can be found by subtracting the coordinates.
For the \(x -\)coordinate: \(x'=x + h\), so \(-6=2 + h\), then \(h=-6 - 2=-8\)
For the \(y -\)coordinate: \(y'=y + k\), so \(1 = 3 + k\), then \(k=1 - 3=-2\)
The translation rule is \((x,y)\to(x - 8,y - 2)\)

Step2: Find the coordinates of \(B'\) and \(C'\)

For point \(B(6,6)\):
Using the translation rule \((x,y)\to(x - 8,y - 2)\)
\(x=6-8=-2\), \(y = 6-2 = 4\)
So \(B'(-2,4)\)

For point \(C(7,2)\):
Using the translation rule \((x,y)\to(x - 8,y - 2)\)
\(x=7-8=-1\), \(y=2 - 2=0\)
So \(C'(-1,0)\)

Answer:

a) \(B'(-2,4)\) and \(C'(-1,0)\)