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 \( \triangle A'B'C' \) (Translation)

Translation \( 6 \) units to the right: \((x,y)\to(x + 6,y)\).

• \(A(-4,-1)\to A'(-4+6,-1)=(2,-1)\)
• \(B(-2,-1)\to B'(-2 + 6,-1)=(4,-1)\)
• \(C(-2,-4)\to C'(-2+6,-4)=(4,-4)\)

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

Dilation with scale factor \(2\) and center at origin: \((x,y)\to(2x,2y)\)

• \(A'(2,-1)\to A''(2\times2,2\times(-1))=(4,-2)\)
• \(B'(4,-1)\to B''(2\times4,2\times(-1))=(8,-2)\)
• \(C'(4,-4)\to C''(2\times4,2\times(-4))=(8,-8)\)

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

Reflection across \(x\) - axis: \((x,y)\to(x,-y)\)

• \(A''(4,-2)\to A'''(4,2)\)
• \(B''(8,-2)\to B'''(8,2)\)
• \(C''(8,-8)\to C'''(8,8)\)

Answer:

\(A'''(4,2)\), \(B'''(8,2)\), \(C'''(8,8)\)