Graph △ABC with vertices A(1, 2), B(3, 4), and C(2, 0). Perform the following sequence of transformations: Step 1: Translate △ABC using the vector to obtain △A′B′C′ Step 2: Reflect △A′B′C′ in the y-axis to obtain △A″B″C″ Step 3: Rotate △A″B″C″ through 90° anticlockwise about the origin to obtain △A‴B‴C‴ Step 4: Enlarge △A‴B‴C‴ with a scale factor of k = 2 centred at the origin to obtain △A⁴B⁴C⁴ Show your work clearly, including: The coordinate rule and coordinates for each transformation step A labelled graph showing the original triangle and all four images (△ABC, △A′B′C′, △A″B″C″, △A‴B‴C‴, △A⁴B⁴C⁴) A comparison of the perimeter of △ABC and △A⁴B⁴C⁴

Question

Asked on May 19, 2026Mathematics

This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.

Accepted Answer · 2026-05-19T19:22:06.186Z

Here are the transformations and calculations: Initial vertices: A(1, 2) B(3, 4) C(2, 0) Step 1: Translate △ABC using the vector -4 \\ -3 to obtain △A′B′C′. The coordinate rule for translation by vector a \\ b is (x, y) (x+a, y+b). For this step, the rule is (x, y) (x-4, y-3). A' = (1-4, 2-3) = (-3, -1) B' = (3-4, 4-3) = (-1, 1) C' = (2-4, 0-3) = (-2, -3) 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). A'' = (-(-3), -1) = (3, -1) B'' = (-(-1), 1) = (1, 1) C'' = (-(-2), -3) = (2, -3) 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). A''' = (-(-1), 3) = (1, 3) B''' = (-(1), 1) = (-1, 1) C''' = (-(-3), 2) = (3, 2) Step 4: Enlarge △A‴B‴C‴ with a scale factor of k = 2 centred at the origin to obtain △A⁴B⁴C⁴. The coordinate rule for enlargement with scale factor k centred at the origin is (x, y) (kx, ky). For this step, the rule is (x, y) (2x, 2y). A^4 = (2 · 1, 2 · 3) = (2, 6) B^4 = (2 · (-1), 2 · 1) = (-2, 2) C^4 = (2 · 3, 2 · 2) = (6, 4) Step 5: Graphing the triangles. To graph, plot the following vertices for each triangle on a coordinate plane and connect them: • △ABC: A(1, 2), B(3, 4), C(2, 0) • △A′B′C′: A′(-3, -1), B′(-1, 1), C′(-2, -3) • △A″B″C″: A″(3, -1), B″(1, 1), C″(2, -3) • △A‴B‴C‴: A‴(1, 3), B‴(-1, 1), C‴(3, 2) • △A⁴B⁴C⁴: A⁴(2, 6), B⁴(-2, 2), C⁴(6, 4) Step 6: Calculate the perimeter of △ABC. The distance formula is 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) Perimeter of △ABC = sqrt(8) + sqrt(17) + sqrt(5) Step 7: Calculate the perimeter of △A⁴B⁴

Accepted Answer · 2026-05-19T19:22:06.186Z

Here are the transformations and calculations: Initial vertices: A(1, 2) B(3, 4) C(2, 0) Step 1: Translate △ABC using the vector -4 \\ -3 to obtain △A′B′C′. The coordinate rule for translation by vector a \\ b is (x, y) (x+a, y+b). For this step, the rule is (x, y) (x-4, y-3). A' = (1-4, 2-3) = (-3, -1) B' = (3-4, 4-3) = (-1, 1) C' = (2-4, 0-3) = (-2, -3) 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). A'' = (-(-3), -1) = (3, -1) B'' = (-(-1), 1) = (1, 1) C'' = (-(-2), -3) = (2, -3) 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). A''' = (-(-1), 3) = (1, 3) B''' = (-(1), 1) = (-1, 1) C''' = (-(-3), 2) = (3, 2) Step 4: Enlarge △A‴B‴C‴ with a scale factor of k = 2 centred at the origin to obtain △A⁴B⁴C⁴. The coordinate rule for enlargement with scale factor k centred at the origin is (x, y) (kx, ky). For this step, the rule is (x, y) (2x, 2y). A^4 = (2 · 1, 2 · 3) = (2, 6) B^4 = (2 · (-1), 2 · 1) = (-2, 2) C^4 = (2 · 3, 2 · 2) = (6, 4) Step 5: Graphing the triangles. To graph, plot the following vertices for each triangle on a coordinate plane and connect them: • △ABC: A(1, 2), B(3, 4), C(2, 0) • △A′B′C′: A′(-3, -1), B′(-1, 1), C′(-2, -3) • △A″B″C″: A″(3, -1), B″(1, 1), C″(2, -3) • △A‴B‴C‴: A‴(1, 3), B‴(-1, 1), C‴(3, 2) • △A⁴B⁴C⁴: A⁴(2, 6), B⁴(-2, 2), C⁴(6, 4) Step 6: Calculate the perimeter of △ABC. The distance formula is 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) Perimeter of △ABC = sqrt(8) + sqrt(17) + sqrt(5) Step 7: Calculate the perimeter of △A⁴B⁴

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