Question

10 triangle abc has coordinates a(-3,-7), b(-3,-3), and c(0,-3). on the graph below, graph and label △abc. graph and state the coordinates of △abc, the image of △abc after a point reflection in the origin. graph and state the coordinates of △abc, the image of △abc reflected in the line y = 2. graph and state the coordinates of △abc, the image of △abc after translation t_{(-8,2)}.

Explanation:

Step1: Reflect over origin (A', B', C')

For a point \((x, y)\), reflection over origin is \((-x, -y)\).

• \(A(-3, -7)\) becomes \(A'(3, 7)\)
• \(B(-3, -3)\) becomes \(B'(3, 3)\)
• \(C(0, -3)\) becomes \(C'(0, 3)\)

Step2: Reflect over \(y = 2\) (A'', B'', C'')

For a point \((x, y)\), reflection over \(y = k\) is \((x, 2k - y)\). Here \(k = 2\), so \(2k - y = 4 - y\).

• \(A'(3, 7)\): \(4 - 7 = -3\) → \(A''(3, -3)\)
• \(B'(3, 3)\): \(4 - 3 = 1\) → \(B''(3, 1)\)
• \(C'(0, 3)\): \(4 - 3 = 1\) → \(C''(0, 1)\)

Step3: Translate \(T_{(-8, 2)}\) (A''', B''', C''')

Translation \((x, y) \to (x - 8, y + 2)\).

• \(A''(3, -3)\): \(3 - 8 = -5\), \(-3 + 2 = -1\) → \(A'''(-5, -1)\)
• \(B''(3, 1)\): \(3 - 8 = -5\), \(1 + 2 = 3\) → \(B'''(-5, 3)\)
• \(C''(0, 1)\): \(0 - 8 = -8\), \(1 + 2 = 3\) → \(C'''(-8, 3)\)

Answer:

• \(\triangle A'B'C'\): \(A'(3, 7)\), \(B'(3, 3)\), \(C'(0, 3)\)
• \(\triangle A''B''C''\): \(A''(3, -3)\), \(B''(3, 1)\), \(C''(0, 1)\)
• \(\triangle A'''B'''C'''\): \(A'''(-5, -1)\), \(B'''(-5, 3)\), \(C'''(-8, 3)\)

(Graphing: Plot original points \(A(-3,-7)\), \(B(-3,-3)\), \(C(0,-3)\); then plot each transformed set using the coordinates above.)