Transformation Properties

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Answer Key & Explanations

Transformation Properties · Grade 8 · Worksheet 1

1. A rectangular solar panel has a length of 3.6 × 10^2 centimeters and a width of 2.5 × 10^1 centimeters. When installed, it is enlarged using a scale factor of 4. What is the area of the enlarged solar panel in square centimeters? Express your answer in scientific notation. Answer: 1.44 × 10^5 Solution: Find the new length after scaling: Original length = 3.6 × 10^2 cm, Scale factor = 4, New length = (3.6 × 10^2) × 4 = 14.4 × 10^2 = 1.44 × 10^3 cm Find the new width after scaling: Original width = 2.5 × 10^1 cm, Scale factor = 4, New width = (2.5 × 10^1) × 4 = 10 × 10^1 = 1.0 × 10^2 cm…
   Full step-by-step solution

   Step 1: Find the new length after scaling: Original length = 3.6 × 10^2 cm, Scale factor = 4, New length = (3.6 × 10^2) × 4 = 14.4 × 10^2 = 1.44 × 10^3 cm Step 2: Find the new width after scaling: Original width = 2.5 × 10^1 cm, Scale factor = 4, New width = (2.5 × 10^1) × 4 = 10 × 10^1 = 1.0 × 10^2 cm Step 3: Calculate the area of the enlarged panel: Area = length × width = (1.44 × 10^3) × (1.0 × 10^2) = 1.44 × 1.0 × 10^(3+2) = 1.44 × 10^5 cm² The answer is 1.44 × 10^5.

2. A triangular garden is drawn on a coordinate plane with vertices at A(2, 3), B(6, 3), and C(4, 7). The triangle is first reflected across the x-axis, then translated 4 units to the left. What are the coordinates of vertex C after both transformations? Answer: (0,-7) Solution: Start with the original coordinates of vertex C: (4, 7) Apply reflection across the x-axis. Reflection across the x-axis changes the sign of the y-coordinate: (4, 7) → (4, -7) Apply translation 4 units to the left.
   Full step-by-step solution

   Step 1: Start with the original coordinates of vertex C: (4, 7) Step 2: Apply reflection across the x-axis. Reflection across the x-axis changes the sign of the y-coordinate: (4, 7) → (4, -7) Step 3: Apply translation 4 units to the left. Moving left decreases the x-coordinate by 4: (4, -7) → (4 - 4, -7) = (0, -7) Step 4: The final coordinates after both transformations are (0, -7)

3. A scientist is studying bacteria growth in a lab. The initial population is 500 bacteria, and it doubles every 3 hours. The scientist uses the exponential growth formula P = 500 × 2^(t/3), where P is the population and t is time in hours. After how many hours will the bacteria population reach 8,000? Answer: 12 Solution: Set up the equation: 500 × 2^(t/3) = 8000 Divide both sides by 500: 2^(t/3) = 16 Recognize that 16 is 2^4, so: 2^(t/3) = 2^4 Since the bases are equal, set the exponents equal: t/3 = 4 Multiply both sides by 3: t = 12 Check: 500 × 2^(12/3) = 500 × 2^4 = 500 × 16 = 8000 The answer is 12 hours.
   Full step-by-step solution

   Step 1: Set up the equation: 500 × 2^(t/3) = 8000 Step 2: Divide both sides by 500: 2^(t/3) = 16 Step 3: Recognize that 16 is 2^4, so: 2^(t/3) = 2^4 Step 4: Since the bases are equal, set the exponents equal: t/3 = 4 Step 5: Multiply both sides by 3: t = 12 Step 6: Check: 500 × 2^(12/3) = 500 × 2^4 = 500 × 16 = 8000 The answer is 12 hours.

4. A triangle has vertices at A(8, 12), B(16, 12), and C(12, 20). After a dilation centered at the origin with a scale factor of 0.5, followed by a reflection over the y-axis, which properties (angles, distances, parallelism) are preserved? Answer: Angles and parallelism are preserved; distances are not preserved. Solution: Identify the transformations. First, a dilation with scale factor 0.5 (centered at origin) reduces all distances by half. Second, a reflection over the y-axis flips the figure horizontally.
   Full step-by-step solution

   Step 1: Identify the transformations. First, a dilation with scale factor 0.5 (centered at origin) reduces all distances by half. Second, a reflection over the y-axis flips the figure horizontally. Step 2: Check distances. Dilation multiplies all side lengths by 0.5, so distances change. Reflection does not change distances, but since dilation already changed them, the final distances are not the same as the original. Therefore, distances are NOT preserved. Step 3: Check angles. Dilation preserves angles (similar figures). Reflection also preserves angles. So angles are preserved. Step 4: Check parallelism. Dilation preserves parallelism (parallel lines remain parallel). Reflection also preserves parallelism. So parallelism is preserved. The answer is: angles and parallelism are preserved; distances are not preserved.

5. Mason is designing a triangular park sign for a hiking trail. The original triangle has side lengths of 15 feet, 20 feet, and 25 feet, with angles measuring approximately 37°, 53°, and 90°. He first reflects the triangle across a vertical line, then dilates the reflected triangle by a scale factor of 2.5. After these two transformations, are all angles and side lengths preserved? Explain which properties are preserved and which are not, and give the final side lengths and angle measures of the dilated triangle. Answer: Angles preserved: 37°, 53°, 90°; side lengths after dilation: 37.5 ft, 50 ft, 62.5 ft Solution: Identify the two transformations: reflection and dilation by 2.5. Reflection preserves all properties: angles, side lengths, and parallelism.
   Full step-by-step solution

   Step 1: Identify the two transformations: reflection and dilation by 2.5. Step 2: Reflection preserves all properties: angles, side lengths, and parallelism. After reflection, the triangle still has side lengths 15 ft, 20 ft, 25 ft and angles 37°, 53°, 90°. Step 3: Dilation multiplies all side lengths by the scale factor (2.5) but preserves angles and parallelism. Step 4: Calculate new side lengths: - 15 ft × 2.5 = 37.5 ft - 20 ft × 2.5 = 50 ft - 25 ft × 2.5 = 62.5 ft Step 5: Angles remain unchanged: 37°, 53°, 90°. Final answer: Angles preserved at 37°, 53°, 90°. Side lengths are not preserved; they become 37.5 ft, 50 ft, and 62.5 ft.

6. A triangle has vertices at (1, 3), (5, 7), and (9, 3). After a dilation with center at the origin and scale factor 3, which properties are preserved? Check: angles, distances, and parallelism. Answer: Angles and parallelism are preserved; distances are not preserved. Solution: Check distances: Original side lengths: between (1,3) and (5,7) = sqrt((5-1)^2 + (7-3)^2) = sqrt(16+16) = sqrt(32) ≈ 5.66.
   Full step-by-step solution

   Step 1: Apply the dilation with scale factor 3 to each vertex: (1,3) -> (3,9), (5,7) -> (15,21), (9,3) -> (27,9). Step 2: Check distances: Original side lengths: between (1,3) and (5,7) = sqrt((5-1)^2 + (7-3)^2) = sqrt(16+16) = sqrt(32) ≈ 5.66. After dilation: between (3,9) and (15,21) = sqrt((15-3)^2 + (21-9)^2) = sqrt(144+144) = sqrt(288) ≈ 16.97. The new distance is 3 times the original, so distances are NOT preserved (they are multiplied by the scale factor). Step 3: Check angles: The shape is similar to the original (same shape, different size). All angle measures remain the same. So angles ARE preserved. Step 4: Check parallelism: Parallel lines remain parallel after dilation because the transformation is a scaling from the origin, which preserves direction. So parallelism IS preserved. The answer is: Angles and parallelism are preserved; distances are not preserved.

7. A triangle with vertices at (7, 9), (12, 9), and (7, 15) is dilated by a factor of 2 with center at the origin. Which properties are preserved: side lengths, angle measures, or both? Answer: angle measures Solution: Identify the original triangle's side lengths. Between (7,9) and (12,9): horizontal distance = 12 - 7 = 5 units. Between (7,9) and (7,15): vertical distance = 15 - 9 = 6 units.
   Full step-by-step solution

   Step 1: Identify the original triangle's side lengths. Between (7,9) and (12,9): horizontal distance = 12 - 7 = 5 units. Between (7,9) and (7,15): vertical distance = 15 - 9 = 6 units. Between (12,9) and (7,15): use distance formula sqrt((12-7)^2 + (9-15)^2) = sqrt(5^2 + (-6)^2) = sqrt(25 + 36) = sqrt(61) ≈ 7.81 units. Step 2: Apply dilation by factor 2. New vertices: (14, 18), (24, 18), (14, 30). New side lengths: between (14,18) and (24,18): 10 units. Between (14,18) and (14,30): 12 units. Between (24,18) and (14,30): sqrt((24-14)^2 + (18-30)^2) = sqrt(10^2 + (-12)^2) = sqrt(100 + 144) = sqrt(244) ≈ 15.62 units. Step 3: Compare side lengths. Original: 5, 6, sqrt(61). New: 10, 12, sqrt(244). Since sqrt(244) = 2*sqrt(61), all sides doubled. Side lengths are NOT preserved (they changed). Step 4: Check angle measures. The triangle is a right triangle (horizontal and vertical sides form a 90-degree angle at (7,9)). After dilation, the sides are still horizontal and vertical, so the 90-degree angle is preserved. The other angles also remain the same because dilation creates a similar triangle. So angle measures ARE preserved. Step 5: Conclusion. Only angle measures are preserved; side lengths are not. The answer is angle measures.