Translate ABC using the vector -3 1 to obtain A'B'C'.

Here are the steps for the composite transformation: Step 1: Translate ABC using the vector -3 \\ 1 to obtain A'B'C'. The coordinate rule for this translation is (x, y) (x-3, y+1). Applying the rule to the vertices A(1, 2), B(3, 4), and C(2, 0): A' = (1-3, 2+1) = (-2, 3) B' = (3-3, 4+1) = (0, 5) C' = (2-3, 0+1) = (-1, 1) The coordinates of A'B'C' are A'(-2, 3), B'(0, 5), and C'(-1, 1). Step 2: Reflect A'B'C' in the y-axis to obtain A''B''C''. The coordinate rule for reflection in the y-axis is (x, y) (-x, y). Applying the rule to the vertices A'(-2, 3), B'(0, 5), and C'(-1, 1): A'' = (-(-2), 3) = (2, 3) B'' = (-(0), 5) = (0, 5) C'' = (-(-1), 1) = (1, 1) The coordinates of A''B''C'' are A''(2, 3), B''(0, 5), and C''(1, 1). Step 3: Rotate A''B''C'' through 90^ anticlockwise about the origin to obtain A'''B'''C'''. The coordinate rule for a 90^ anticlockwise rotation about the origin is (x, y) (-y, x). Applying the rule to the vertices A''(2, 3), B''(0, 5), and C''(1, 1): A''' = (-(3), 2) = (-3, 2) B''' = (-(5), 0) = (-5, 0) C''' = (-(1), 1) = (-1, 1) The coordinates of A'''B'''C''' are A'''(-3, 2), B'''(-5, 0), and C'''(-1, 1). Step 4: Enlarge A'''B'''C''' with a scale factor of k=2 centred at the origin to obtain A''''B''''C''''. The coordinate rule for enlargement with scale factor k=2 centred at the origin is (x, y) (2x, 2y). Applying the rule to the vertices A'''(-3, 2), B'''(-5, 0), and C'''(-1, 1): A'''' = (2 × -3, 2 × 2) = (-6, 4) B'''' = (2 × -5, 2 × 0) = (-10, 0) C'''' = (2 × -1, 2 × 1) = (-2, 2) The coordinates of A''''B''''C'''' are A''''(-6, 4), B''''(-10, 0), and C''''(-2, 2). To create the labeled graph, plot the original triangle ABC and all the transformed images A'B'C', A''B''C'', A'''B'''C''', and A''''B''''C'''' using the coordinates calculated above on a coordinate plane. Step 5: Compare the perimeter of ABC and A''''B''''C''''. First, calculate the side lengths of ABC using the distance formula d = sqrt((x_2-x_1)^2 + (y_2-y_1)^2): AB = sqrt((3-1)^2 + (4-2)^2) = sqrt(2^2 + 2^2) = sqrt(4+4) = sqrt(8) BC = sqrt((2-3)^2 + (0-4)^2) = sqrt((-1)^2 + (-4)^2) = sqrt(1+16) = sqrt(17) CA = sqrt((1-2)^2 + (2-0)^2) = sqrt((-1)^2 + 2^2) = sqrt(1+4) = sqrt(5) The perimeter of ABC is P_ABC = AB + BC + CA = sqrt(8) + sqrt(17) + sqrt(5). The final transformation from A'''B'''C''' to A''''B''''C'''' is an enlargement with a scale factor of k=2. All previous transformations (translation, reflection, rotation) are rigid transformations, meaning they preserve length and shape. Therefore, A'''B'''C''' has the same perimeter as ABC. When a shape is enlarged by a scale factor k, its perimeter is multiplied by k. So, the perimeter of A''''B''''C'''' is k times the perimeter of ABC. P_A''''B