Transformation Sequences

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Answer Key & Explanations

Transformation Sequences · Grade 8 · Worksheet 3

1. 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 translated 2 units right and 3 units down; then, it is reflected across the y-axis; finally, it is rotated 180° counterclockwise about the origin. What are the coordinates of the final image of vertex A? Answer: (1, -1) Solution: Start with original vertex A(1, 2). Apply translation 2 units right and 3 units down: A(1+2, 2-3) = A(3, -1). Apply reflection across the y-axis: A(-3, -1).
   Full step-by-step solution

   Step 1: Start with original vertex A(1, 2). Step 2: Apply translation 2 units right and 3 units down: A(1+2, 2-3) = A(3, -1). Step 3: Apply reflection across the y-axis: A(-3, -1). Step 4: Apply 180° counterclockwise rotation about the origin: A(1, 1). The final coordinates of vertex A are (1, 1).

2. Noah is creating a digital art piece using a coordinate plane. He starts with a triangle that has vertices at A(7, 12), B(13, 12), and C(10, 18). First, he dilates the triangle by a scale factor of 1/2 using the origin as the center of dilation. Then, he rotates the dilated triangle 90 degrees clockwise about the origin. Finally, he translates the rotated triangle 9 units to the left and 5 units down. What are the coordinates of vertex A after this sequence of transformations? Answer: (3, 1) Solution: Start with original point A(7, 12). Apply dilation with a scale factor of 1/2 about the origin: (x, y) -> (x/2, y/2). A(7, 12) -> A'(3.5, 6).
   Full step-by-step solution

   Step 1: Start with original point A(7, 12). Step 2: Apply dilation with a scale factor of 1/2 about the origin: (x, y) -> (x/2, y/2). A(7, 12) -> A'(3.5, 6). Step 3: Apply a 90-degree clockwise rotation about the origin: (x, y) -> (y, -x). A'(3.5, 6) -> A''(6, -3.5). Step 4: Apply translation 9 units left and 5 units down: (x, y) -> (x - 9, y - 5). A''(6, -3.5) -> A'''(6 - 9, -3.5 - 5) = (-3, -8.5). The final coordinates of vertex A are (-3, -8.5).

3. Liam is designing a logo that starts as a triangle with vertices at (2, 1), (4, 1), and (3, 4). He first translates the triangle 3 units left and 2 units up. Then he reflects the translated triangle across the y-axis. Finally, he dilates the reflected triangle by a scale factor of 2 with the origin as the center of dilation. What are the coordinates of the final triangle's vertices? Answer: (-2, 6), (-6, 6), (-4, 12) Solution: Geometric transformations follow specific rules: translations shift all points by fixed amounts, reflections flip shapes across an axis (changing the sign of x-coordinates for y-axis reflection), and dilations scale all distances from a center point.
   Full step-by-step solution

   Geometric transformations follow specific rules: translations shift all points by fixed amounts, reflections flip shapes across an axis (changing the sign of x-coordinates for y-axis reflection), and dilations scale all distances from a center point. The order of transformations matters, and each transformation must be applied to the result of the previous one, not to the original shape.

4. Kaia is designing a geometric logo for her school's sports team. She starts with a triangle with vertices at A(9, 12), B(15, 12), and C(12, 18). She first reflects the triangle across the x-axis. Then she rotates the reflected triangle 90 degrees clockwise about the origin. Finally, she dilates the rotated triangle by a scale factor of 2 with the origin as the center of dilation. What are the coordinates of vertex C after this sequence of transformations? Answer: (36, -24) Solution: Start with original vertex C(12, 18). Reflect across the x-axis. Rule: (x, y) -> (x, -y).
   Full step-by-step solution

   Step 1: Start with original vertex C(12, 18). Step 2: Reflect across the x-axis. Rule: (x, y) -> (x, -y). So (12, 18) becomes (12, -18). Step 3: Rotate 90 degrees clockwise about the origin. Rule: (x, y) -> (y, -x). So (12, -18) becomes (-18, -12). Step 4: Dilate by scale factor 2 with origin as center. Rule: (x, y) -> (2x, 2y). So (-18, -12) becomes (2 * -18, 2 * -12) = (-36, -24). The final coordinates of vertex C are (-36, -24).

5. √(64) + 3² × (2³ - 4) = ? Answer: 44 Solution: Calculate inside the parentheses: 2³ - 4 = 8 - 4 = 4 Calculate the square root: √(64) = 8 Calculate the exponent: 3² = 9 Multiply: 9 × 4 = 36 Add: 8 + 36 = 44 The answer is 44.
   Full step-by-step solution

   Step 1: Calculate inside the parentheses: 2³ - 4 = 8 - 4 = 4 Step 2: Calculate the square root: √(64) = 8 Step 3: Calculate the exponent: 3² = 9 Step 4: Multiply: 9 × 4 = 36 Step 5: Add: 8 + 36 = 44 The answer is 44.

6. Translate 9 units right and 7 units up, then reflect over the y-axis, then rotate 180° clockwise about the origin. Answer: Translate 9 units left and 7 units down, then reflect over the y-axis Solution: Start with the original sequence: Translate 9 units right and 7 units up, then reflect over the y-axis, then rotate 180° clockwise. Work backwards. The last transformation is a 180° clockwise rotation.
   Full step-by-step solution

   Step 1: Start with the original sequence: Translate 9 units right and 7 units up, then reflect over the y-axis, then rotate 180° clockwise. Step 2: Work backwards. The last transformation is a 180° clockwise rotation. Its inverse is a 180° counterclockwise rotation. A 180° rotation (either direction) maps (x, y) to (-x, -y). Step 3: Apply this inverse to undo the rotation: The sequence becomes: Translate 9 units right and 7 units up, then reflect over the y-axis. Step 4: The next transformation to undo is the reflection over the y-axis. Its inverse is itself: reflecting over the y-axis again. A reflection over the y-axis maps (x, y) to (-x, y). Step 5: Apply this inverse: The sequence becomes: Translate 9 units right and 7 units up. Step 6: The next transformation to undo is the translation. Its inverse is translating 9 units left and 7 units down. Step 7: Now, re-read the problem: The question asks for the sequence that would undo the original. So the answer is the inverse sequence we found: Translate 9 units left and 7 units down, then reflect over the y-axis. The answer is: Translate 9 units left and 7 units down, then reflect over the y-axis.

7. (4.2 × 10⁶) ÷ (2.1 × 10²) = ? Answer: 20000 Solution: Divide the coefficients: 4.2 ÷ 2.1 = 2 Subtract the exponents: 6 - 2 = 4 Combine the results: 2 × 10⁴ Convert to standard form: 2 × 10,000 = 20,000 The answer is 20000.
   Full step-by-step solution

   Step 1: Divide the coefficients: 4.2 ÷ 2.1 = 2 Step 2: Subtract the exponents: 6 - 2 = 4 Step 3: Combine the results: 2 × 10⁴ Step 4: Convert to standard form: 2 × 10,000 = 20,000 The answer is 20000.