Worksheet 1Worksheet 2Worksheet 3

The download includes this worksheet followed by the answer key with explanations. In the print dialog choose “Save as PDF” to download.

lessonbunny.com
Name: ______________________________ Date: ______________

Transformation Sequences

Grade 8 · Geometry · Worksheet 2

  1. A rectangular garden has a length of 15 meters and a width of 10 meters. The owner wants to create a path of uniform width around the entire garden, which will increase the total area to 266 square meters. What is the width of the path in meters?
    Answer: ______________
  2. A rectangle on a coordinate plane has 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. (4² × 3) - (√144 ÷ 2) = ? Answer: ______________
  4. Translate 6 units left and 1 unit down, then reflect over the y-axis, then rotate 90° clockwise about the origin. Answer: ______________
  5. (5.4 × 10⁷) ÷ (1.8 × 10⁴) = ? Answer: ______________
  6. (3² × 4) - (2³ ÷ 2) = ? Answer: ______________
  7. Translate 12 units right and 15 units up, then reflect over the y-axis, then rotate 180° clockwise about the origin. Answer: ______________
lessonbunny.com

Answer Key & Explanations

Transformation Sequences · Grade 8 · Worksheet 2

  1. A rectangular garden has a length of 15 meters and a width of 10 meters. The owner wants to create a path of uniform width around the entire garden, which will increase the total area to 266 square meters. What is the width of the path in meters? Answer: 2 Solution: Let x be the width of the path in meters. The path adds 2x to both the length and width (x on each side).
    Full step-by-step solution

    Step 1: Let x be the width of the path in meters. Step 2: The path adds 2x to both the length and width (x on each side). Step 3: New length = 15 + 2x Step 4: New width = 10 + 2x Step 5: Total area with path = (15 + 2x)(10 + 2x) = 266 Step 6: Expand: 150 + 30x + 20x + 4x^2 = 266 Step 7: Simplify: 4x^2 + 50x + 150 = 266 Step 8: Subtract 266: 4x^2 + 50x - 116 = 0 Step 9: Divide by 2: 2x^2 + 25x - 58 = 0 Step 10: Factor: (2x + 29)(x - 2) = 0 Step 11: Solutions: x = -29/2 or x = 2 Step 12: Only x = 2 makes sense (width cannot be negative). The width of the path is 2 meters.

  2. A rectangle on a coordinate plane has 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 vertex A at (1, 2). Rotate 90° counterclockwise about the origin. The rule for this rotation is (x, y) → (-y, x).
    Full step-by-step solution

    Step 1: Start with vertex A at (1, 2). Step 2: Rotate 90° counterclockwise about the origin. The rule for this rotation is (x, y) → (-y, x). So (1, 2) becomes (-2, 1). Step 3: Reflect across the y-axis. The rule for this reflection is (x, y) → (-x, y). So (-2, 1) becomes (2, 1). Step 4: Translate 2 units up. This adds 2 to the y-coordinate. So (2, 1) becomes (2, 3). The final coordinates of vertex A after all transformations are (2, 3).

  3. (4² × 3) - (√144 ÷ 2) = ? Answer: 42 Solution: Calculate inside the first parentheses: 4² × 3 4² = 16 16 × 3 = 48 Calculate inside the second parentheses: √144 ÷ 2 √144 = 12 12 ÷ 2 = 6 Perform the subtraction: 48 - 6 = 42 The answer is 42.
    Full step-by-step solution

    Step 1: Calculate inside the first parentheses: 4² × 3 4² = 16 16 × 3 = 48 Step 2: Calculate inside the second parentheses: √144 ÷ 2 √144 = 12 12 ÷ 2 = 6 Step 3: Perform the subtraction: 48 - 6 = 42 The answer is 42.

  4. Translate 6 units left and 1 unit down, then reflect over the y-axis, then rotate 90° clockwise about the origin. Answer: Translate 1 unit up and 6 units right, then reflect over the x-axis Solution: Start with the original sequence: Translate 6 left and 1 down, then reflect over y-axis, then rotate 90° clockwise.
    Full step-by-step solution

    Step 1: Start with the original sequence: Translate 6 left and 1 down, then reflect over y-axis, then rotate 90° clockwise. Step 2: Apply the inverse of the last transformation first. The inverse of a 90° clockwise rotation is a 90° counterclockwise rotation, which maps (x, y) to (-y, x). Step 3: Apply this to our sequence: The rotation undoes itself, leaving: Translate 6 left and 1 down, then reflect over y-axis. Step 4: A reflection over the y-axis maps (x, y) to (-x, y). Step 5: Combining the translation and reflection: Translating 6 left and 1 down gives (x-6, y-1), then reflecting over y-axis gives (-(x-6), y-1) = (-x+6, y-1). Step 6: This is equivalent to reflecting over x-axis first: (x, -y), then translating 1 up and 6 right: (x+6, -y+1). Step 7: Therefore, the equivalent single transformation is: Reflect over x-axis, then translate 1 up and 6 right.

  5. (5.4 × 10⁷) ÷ (1.8 × 10⁴) = ? Answer: 3000 Solution: Write the expression as (5.4 ÷ 1.8) × (10⁷ ÷ 10⁴). Calculate 5.4 ÷ 1.8 = 3. Calculate 10⁷ ÷ 10⁴ = 10^(7-4) = 10³.
    Full step-by-step solution

    Step 1: Write the expression as (5.4 ÷ 1.8) × (10⁷ ÷ 10⁴). Step 2: Calculate 5.4 ÷ 1.8 = 3. Step 3: Calculate 10⁷ ÷ 10⁴ = 10^(7-4) = 10³. Step 4: Combine the results: 3 × 10³ = 3000. The answer is 3000.

  6. (3² × 4) - (2³ ÷ 2) = ? Answer: 32 Solution: We have: (3² × 4) - (2³ ÷ 2) = ?
    Full step-by-step solution

    Let's solve step-by-step. We have: (3² × 4) - (2³ ÷ 2) = ? **Step 1: Handle exponents first (order of operations)** 3² = 3 × 3 = 9 2³ = 2 × 2 × 2 = 8 So the expression becomes: (9 × 4) - (8 ÷ 2) **Step 2: Perform multiplication and division inside parentheses** 9 × 4 = 36 8 ÷ 2 = 4 Now we have: 36 - 4 **Step 3: Perform subtraction** 36 - 4 = 32 **Final Answer:** 32

  7. Translate 12 units right and 15 units up, then reflect over the y-axis, then rotate 180° clockwise about the origin. Answer: Translate 12 units left and 15 units down, then reflect over the y-axis Solution: Start with the original sequence: Translate 12 right and 15 up, then reflect over y-axis, then rotate 180° clockwise.
    Full step-by-step solution

    Step 1: Start with the original sequence: Translate 12 right and 15 up, then reflect over y-axis, then rotate 180° clockwise. Step 2: Apply the inverse of the last transformation first. The inverse of a 180° clockwise rotation is a 180° counterclockwise rotation (which is the same as 180° clockwise), so it maps (x, y) to (-x, -y). Step 3: This undoes the rotation, leaving: Translate 12 right and 15 up, then reflect over y-axis. Step 4: A reflection over the y-axis maps (x, y) to (-x, y). Step 5: Combining the translation and reflection: Translating 12 right and 15 up gives (x+12, y+15), then reflecting over y-axis gives (-(x+12), y+15) = (-x-12, y+15). Step 6: This is equivalent to reflecting over y-axis first: (-x, y), then translating 12 left and 15 up: (-x-12, y+15). Step 7: Therefore, the equivalent single transformation is: Reflect over y-axis, then translate 12 left and 15 up.