Question

erika translates \\( \triangle a b c \\) 6 units to the right to form \\( \triangle a ^ { prime } b ^ { prime } c ^ { prime } \\). then she dilates \\( \triangle a ^ { prime } b ^ { prime } c ^ { prime } \\) by a scale factor of 2 with the origin as the center of dilation to form \\( \triangle a ^ { prime prime } b ^ { prime prime } c ^ { prime prime } \\). lastly, she reflects \\( \triangle a ^ { prime prime } b ^ { prime prime } c ^ { prime prime } \\) across the \\( x \\)-axis to form \\( \triangle a ^ { prime prime prime } b ^ { prime prime prime } c ^ { prime prime prime } \\).
what are the coordinates of \\( \triangle a ^ { prime prime prime } b ^ { prime prime prime } c ^ { prime prime prime } \\)?

Explanation:

Step1: Find coordinates of \(\triangle A'B'C'\) after translation

Translation rule: \((x,y)\to(x + 6,y)\).
Given \(A(-3,-1)\), \(B(-1,-1)\), \(C(-2,-4)\).
For \(A\): \((-3+6,-1)=(3,-1)\).
For \(B\): \((-1 + 6,-1)=(5,-1)\).
For \(C\): \((-2+6,-4)=(4,-4)\).

Step2: Find coordinates of \(\triangle A''B''C''\) after dilation

Dilation rule (center at origin, scale factor \(k = 2\)): \((x,y)\to(2x,2y)\).
For \(A'\)(\(3,-1\)): \((2\times3,2\times(-1))=(6,-2)\).
For \(B'\)(\(5,-1\)): \((2\times5,2\times(-1))=(10,-2)\).
For \(C'\)(\(4,-4\)): \((2\times4,2\times(-4))=(8,-8)\).

Step3: Find coordinates of \(\triangle A'''B'''C'''\) after reflection

Reflection across \(x -\)axis rule: \((x,y)\to(x,-y)\).
For \(A''\)(\(6,-2\)): \((6,2)\).
For \(B''\)(\(10,-2\)): \((10,2)\).
For \(C''\)(\(8,-8\)): \((8,8)\).

Answer:

\(A'''(6,2)\), \(B'''(10,2)\), \(C'''(8,8)\)