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Transformation Sequences

Grade 8 · Geometry · Worksheet 1

  1. √(169) + 4³ - (5² × 2) = ? Answer: ______________
  2. A rectangle is drawn on a coordinate plane with vertices at A(1, 2), B(4, 2), C(4, 5), and D(1, 5). It undergoes the following sequence of transformations: first, it is rotated 90° counterclockwise about the origin; then, it is reflected across the y-axis; finally, it is translated 2 units up. What are the coordinates of the final image of vertex A? Answer: ______________
  3. Liam is designing a geometric pattern for a mural. He starts with a triangle that has vertices at (2, 3), (4, 7), and (6, 1). He first translates the triangle 5 units left and 2 units up. Then he reflects the resulting shape across the y-axis. Finally, he rotates the figure 90 degrees counterclockwise about the origin. What are the coordinates of the vertices of the final triangle? Answer: ______________
  4. Mason is designing a digital logo for his school club. He starts with a triangle that has vertices at A(2, 7), B(7, 2), and C(7, 7). He first reflects the triangle across the y-axis. Then he rotates the reflected triangle 90° clockwise about the origin. Finally, he translates the rotated triangle 2 units left and 7 units down. What are the coordinates of vertex A after this sequence of transformations? Answer: ______________
  5. Liam is designing a logo that starts with a triangle at coordinates A(2, 3), B(5, 3), C(3, 6). He first reflects the triangle across the y-axis, then translates it 4 units to the right, and finally rotates it 90° counterclockwise about the origin. What are the coordinates of point C after this sequence of transformations? Answer: ______________
  6. A triangle has vertices at A(2, 1), B(5, 1), and C(3, 4) on a coordinate plane. It undergoes the following sequence of transformations: first, it is reflected across the x-axis; then, it is translated 3 units to the left; finally, it is dilated by a scale factor of 2 with the origin as the center of dilation. What are the coordinates of the final image of vertex C? Answer: ______________
  7. (4² - 2³) × (√81 ÷ 3) = ? Answer: ______________
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Answer Key & Explanations

Transformation Sequences · Grade 8 · Worksheet 1

  1. √(169) + 4³ - (5² × 2) = ? Answer: 27 Solution: Calculate the square root: √(169) = 13 Calculate the exponent: 4³ = 64 Calculate inside the parentheses: 5² = 25, then 25 × 2 = 50 Substitute the values: 13 + 64 - 50 Perform addition and subtraction from left to right: 13 + 64 = 77, then 77 - 50 = 27 The answer is 27.
    Full step-by-step solution

    Step 1: Calculate the square root: √(169) = 13 Step 2: Calculate the exponent: 4³ = 64 Step 3: Calculate inside the parentheses: 5² = 25, then 25 × 2 = 50 Step 4: Substitute the values: 13 + 64 - 50 Step 5: Perform addition and subtraction from left to right: 13 + 64 = 77, then 77 - 50 = 27 The answer is 27.

  2. A rectangle is drawn on a coordinate plane with vertices at A(1, 2), B(4, 2), C(4, 5), and D(1, 5). It undergoes the following sequence of transformations: first, it is rotated 90° counterclockwise about the origin; then, it is reflected across the y-axis; finally, it is translated 2 units up. What are the coordinates of the final image of vertex A? Answer: (2, 3) Solution: Start with original coordinates of vertex A: (1, 2) Apply rotation 90° counterclockwise about the origin. The rule for this rotation is (x, y) → (-y, x). So (1, 2) becomes (-2, 1).
    Full step-by-step solution

    Step 1: Start with original coordinates of vertex A: (1, 2) Step 2: Apply rotation 90° counterclockwise about the origin. The rule for this rotation is (x, y) → (-y, x). So (1, 2) becomes (-2, 1). Step 3: Apply reflection across the y-axis. The rule for this reflection is (x, y) → (-x, y). So (-2, 1) becomes (2, 1). Step 4: Apply translation 2 units up. The rule for this translation is (x, y) → (x, y + 2). So (2, 1) becomes (2, 3). The final coordinates of vertex A after all transformations are (2, 3).

  3. Liam is designing a geometric pattern for a mural. He starts with a triangle that has vertices at (2, 3), (4, 7), and (6, 1). He first translates the triangle 5 units left and 2 units up. Then he reflects the resulting shape across the y-axis. Finally, he rotates the figure 90 degrees counterclockwise about the origin. What are the coordinates of the vertices of the final triangle? Answer: (-5, -4), (-9, -6), (-3, -8) Solution: Geometric transformations follow specific rules: translation shifts all points by the same amount, reflection creates a mirror image across a line, and rotation turns the figure around a fixed point.
    Full step-by-step solution

    Geometric transformations follow specific rules: translation shifts all points by the same amount, reflection creates a mirror image across a line, and rotation turns the figure around a fixed point. The order of transformations matters because performing them in different sequences can produce different final results. When working with multiple transformations, it's helpful to apply them one at a time to track how the figure moves through each step.

  4. Mason is designing a digital logo for his school club. He starts with a triangle that has vertices at A(2, 7), B(7, 2), and C(7, 7). He first reflects the triangle across the y-axis. Then he rotates the reflected triangle 90° clockwise about the origin. Finally, he translates the rotated triangle 2 units left and 7 units down. What are the coordinates of vertex A after this sequence of transformations? Answer: (-5, -9) Solution: Start with original point A(2, 7). Reflect across the y-axis. Rule: (x, y) → (-x, y).
    Full step-by-step solution

    Step 1: Start with original point A(2, 7). Step 2: Reflect across the y-axis. Rule: (x, y) → (-x, y). So (2, 7) → (-2, 7). Step 3: Rotate 90° clockwise about the origin. Rule: (x, y) → (y, -x). So (-2, 7) → (7, -(-2)) = (7, 2). Step 4: Translate 2 units left and 7 units down. Rule: (x, y) → (x - 2, y - 7). So (7, 2) → (7 - 2, 2 - 7) = (5, -5). The final coordinates of vertex A are (5, -5).

  5. Liam is designing a logo that starts with a triangle at coordinates A(2, 3), B(5, 3), C(3, 6). He first reflects the triangle across the y-axis, then translates it 4 units to the right, and finally rotates it 90° counterclockwise about the origin. What are the coordinates of point C after this sequence of transformations? Answer: (1, -7) Solution: Geometric transformations follow specific rules. Reflection across the y-axis changes (x, y) to (-x, y). A 90° counterclockwise rotation about the origin transforms (x, y) to (-y, x).
    Full step-by-step solution

    Geometric transformations follow specific rules. Reflection across the y-axis changes (x, y) to (-x, y). Translation moves points by adding to their coordinates. A 90° counterclockwise rotation about the origin transforms (x, y) to (-y, x). When applying multiple transformations, you must perform them in the given order, as changing the order can produce different results. This concept is important in computer graphics, animation, and engineering design.

  6. A triangle has vertices at A(2, 1), B(5, 1), and C(3, 4) on a coordinate plane. It undergoes the following sequence of transformations: first, it is reflected across the x-axis; then, it is translated 3 units to the left; finally, it is dilated by a scale factor of 2 with the origin as the center of dilation. What are the coordinates of the final image of vertex C? Answer: (0, -8) Solution: When reflecting across the x-axis, the x-coordinate stays the same, and the y-coordinate changes sign. So, C(3, 4) becomes C'(3, -4).
    Full step-by-step solution

    Let's go step-by-step for vertex C(3, 4). **Step 1: Reflection across the x-axis** When reflecting across the x-axis, the x-coordinate stays the same, and the y-coordinate changes sign. So, C(3, 4) becomes C'(3, -4). **Step 2: Translation 3 units to the left** Translating left means subtract 3 from the x-coordinate; y-coordinate stays the same. C'(3, -4) → C''(3 - 3, -4) = C''(0, -4). **Step 3: Dilation by scale factor 2 with origin as center** Multiply both coordinates by 2. C''(0, -4) → C'''(0 × 2, -4 × 2) = C'''(0, -8). **Final Answer:** (0, -8)

  7. (4² - 2³) × (√81 ÷ 3) = ? Answer: 24 Solution: Calculate inside the first parentheses: 4² = 16 and 2³ = 8, so 16 - 8 = 8 Calculate inside the second parentheses: √81 = 9, then 9 ÷ 3 = 3 Multiply the results: 8 × 3 = 24 The answer is 24.
    Full step-by-step solution

    Step 1: Calculate inside the first parentheses: 4² = 16 and 2³ = 8, so 16 - 8 = 8 Step 2: Calculate inside the second parentheses: √81 = 9, then 9 ÷ 3 = 3 Step 3: Multiply the results: 8 × 3 = 24 The answer is 24.