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Transformation Properties

Grade 8 · Geometry · Worksheet 3

  1. Aroha is an artist creating a design using a triangular piece of stained glass. The triangle has vertices at points A(1, 3), B(5, 3), and C(3, 7). She first reflects the triangle across the x-axis. Then she dilates the reflected triangle by a scale factor of 3, with the origin as the center of dilation. After these two transformations, does the final triangle preserve the original triangle's angle measures and side lengths? Explain your reasoning and state which properties are preserved and which are not. Answer: ______________
  2. A triangle has vertices at (2, 7), (7, 12), and (12, 7). After a dilation with center at the origin and scale factor 2, what properties are preserved? Check if the angles are preserved and if the side lengths are preserved. State your answer as 'angles preserved, distances not preserved' or 'angles not preserved, distances preserved' or 'both preserved' or 'neither preserved'. Answer: ______________
  3. A rectangular swimming pool is being designed for a community center. The pool's length is 25 meters and its width is 15 meters. The designers want to create a scale model where 1 centimeter represents 2.5 meters. What will be the perimeter of the pool in centimeters on the scale model?
    Answer: ______________
  4. Emma has a triangular garden with side lengths of 7 meters, 9 meters, and 11 meters. She wants to create a scale model of the garden using a dilation with a scale factor of 3. Which properties of the original triangle are preserved in the dilated triangle? Specifically, are the side lengths and angle measures preserved or changed? Explain your reasoning. Answer: ______________
  5. Aroha applies a dilation with center at the origin and scale factor 3 to triangle ABC with vertices A(1, 2), B(3, 5), and C(7, 1). She then reflects the dilated triangle over the y-axis. Which properties are preserved from the original triangle ABC to the final triangle after both transformations? Check all that apply: angle measures, side lengths, parallelism, and orientation. Answer: ______________
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Answer Key & Explanations

Transformation Properties · Grade 8 · Worksheet 3

  1. Aroha is an artist creating a design using a triangular piece of stained glass. The triangle has vertices at points A(1, 3), B(5, 3), and C(3, 7). She first reflects the triangle across the x-axis. Then she dilates the reflected triangle by a scale factor of 3, with the origin as the center of dilation. After these two transformations, does the final triangle preserve the original triangle's angle measures and side lengths? Explain your reasoning and state which properties are preserved and which are not. Answer: Angle measures are preserved; side lengths are not preserved. Solution: Reflect triangle ABC across the x-axis. Reflection is a rigid transformation, so it preserves both angle measures and side lengths. Dilate the reflected triangle by a scale factor of 3 with the origin as the center.
    Full step-by-step solution

    Step 1: Reflect triangle ABC across the x-axis. Reflection is a rigid transformation, so it preserves both angle measures and side lengths. The reflected triangle has the same shape and size as the original. Step 2: Dilate the reflected triangle by a scale factor of 3 with the origin as the center. Dilation preserves angle measures but changes side lengths by multiplying each side by the scale factor. Here, all side lengths become 3 times longer. Step 3: Analyze the combined effect. Since reflection preserved angles and dilation also preserves angles, the final triangle has the same angle measures as the original. However, dilation multiplies side lengths by 3, so the side lengths are not preserved (they are 3 times the original). Final answer: Angle measures are preserved, but side lengths are not preserved.

  2. A triangle has vertices at (2, 7), (7, 12), and (12, 7). After a dilation with center at the origin and scale factor 2, what properties are preserved? Check if the angles are preserved and if the side lengths are preserved. State your answer as 'angles preserved, distances not preserved' or 'angles not preserved, distances preserved' or 'both preserved' or 'neither preserved'. Answer: angles preserved, distances not preserved Solution: Find the coordinates of the dilated triangle. Multiply each coordinate by the scale factor 2: (2,7) -> (4,14), (7,12) -> (14,24), (12,7) -> (24,14). Check if angles are preserved.
    Full step-by-step solution

    Step 1: Find the coordinates of the dilated triangle. Multiply each coordinate by the scale factor 2: (2,7) -> (4,14), (7,12) -> (14,24), (12,7) -> (24,14). Step 2: Check if angles are preserved. In a dilation, the shape is similar to the original, so all angles remain the same. For example, the original triangle has a right angle at (2,7) because the sides are horizontal and vertical. The dilated triangle also has a right angle at (4,14). So angles are preserved. Step 3: Check if side lengths are preserved. Original side lengths: between (2,7) and (7,12) is sqrt((7-2)^2 + (12-7)^2) = sqrt(25+25) = sqrt(50) ≈ 7.07. Between (7,12) and (12,7) is sqrt((12-7)^2 + (7-12)^2) = sqrt(25+25) = sqrt(50) ≈ 7.07. Between (2,7) and (12,7) is 10. Dilated side lengths: between (4,14) and (14,24) is sqrt((14-4)^2 + (24-14)^2) = sqrt(100+100) = sqrt(200) ≈ 14.14. Between (14,24) and (24,14) is sqrt((24-14)^2 + (14-24)^2) = sqrt(100+100) = sqrt(200) ≈ 14.14. Between (4,14) and (24,14) is 20. The side lengths are doubled, not preserved. Step 4: Conclusion: Angles are preserved, but distances (side lengths) are not preserved. The answer is 'angles preserved, distances not preserved'.

  3. A rectangular swimming pool is being designed for a community center. The pool's length is 25 meters and its width is 15 meters. The designers want to create a scale model where 1 centimeter represents 2.5 meters. What will be the perimeter of the pool in centimeters on the scale model? Answer: 32 Solution: Actual perimeter = 2 × (length + width) = 2 × (25 + 15) = 2 × 40 = 80 meters The scale is 1 cm : 2.5 m, which means 1 cm on the model represents 2.5 m in reality Scale factor = 1 cm / 2.5 m = 1/2.5 = 0.4 Model perimeter = Actual perimeter × Scale factor = 80 × 0.4 = 32 cm Model length = 25 m ×…
    Full step-by-step solution

    Step 1: Find the actual perimeter of the pool Actual perimeter = 2 × (length + width) = 2 × (25 + 15) = 2 × 40 = 80 meters Step 2: Determine the scale factor The scale is 1 cm : 2.5 m, which means 1 cm on the model represents 2.5 m in reality Scale factor = 1 cm / 2.5 m = 1/2.5 = 0.4 Step 3: Apply the scale factor to the perimeter Model perimeter = Actual perimeter × Scale factor = 80 × 0.4 = 32 cm Step 4: Verify by scaling dimensions first Model length = 25 m × 0.4 = 10 cm Model width = 15 m × 0.4 = 6 cm Model perimeter = 2 × (10 + 6) = 2 × 16 = 32 cm The answer is 32.

  4. Emma has a triangular garden with side lengths of 7 meters, 9 meters, and 11 meters. She wants to create a scale model of the garden using a dilation with a scale factor of 3. Which properties of the original triangle are preserved in the dilated triangle? Specifically, are the side lengths and angle measures preserved or changed? Explain your reasoning. Answer: Angle measures are preserved; side lengths are multiplied by 3 (not preserved as same values). Solution: A dilation with a scale factor of 3 multiplies all side lengths by 3. The original side lengths are 7 m, 9 m, and 11 m. After dilation, the new side lengths are 21 m, 27 m, and 33 m.
    Full step-by-step solution

    Step 1: Understand the transformation. A dilation with a scale factor of 3 multiplies all side lengths by 3. The original side lengths are 7 m, 9 m, and 11 m. After dilation, the new side lengths are 21 m, 27 m, and 33 m. So side lengths are not preserved; they are scaled by a factor of 3. Step 2: Consider angle measures. A dilation is a similarity transformation, meaning it preserves the shape but not the size. The angles of the triangle remain exactly the same because the triangle is enlarged proportionally. So angle measures are preserved. Step 3: Final conclusion. In a dilation, side lengths change (they are multiplied by the scale factor), but angle measures remain unchanged. Therefore, the properties preserved are the angle measures, while side lengths are not preserved. The answer is: Angle measures are preserved; side lengths are multiplied by 3 (not preserved as same values).

  5. Aroha applies a dilation with center at the origin and scale factor 3 to triangle ABC with vertices A(1, 2), B(3, 5), and C(7, 1). She then reflects the dilated triangle over the y-axis. Which properties are preserved from the original triangle ABC to the final triangle after both transformations? Check all that apply: angle measures, side lengths, parallelism, and orientation. Answer: angle measures, parallelism Solution: Dilation with scale factor 3 multiplies all coordinates by 3. The dilated triangle has vertices A'(3, 6), B'(9, 15), C'(21, 3).
    Full step-by-step solution

    Step 1: Dilation with scale factor 3 multiplies all coordinates by 3. The dilated triangle has vertices A'(3, 6), B'(9, 15), C'(21, 3). Dilation preserves angle measures and parallelism, but side lengths are multiplied by 3 (not preserved), and orientation is preserved (no flip). Step 2: Reflection over the y-axis changes the sign of the x-coordinates. The final triangle has vertices A''(-3, 6), B''(-9, 15), C''(-21, 3). Reflection preserves angle measures, side lengths, and parallelism, but flips orientation (clockwise becomes counterclockwise). Step 3: Combine the effects: - Angle measures: Preserved by dilation AND reflection, so preserved overall. - Side lengths: Changed by dilation (multiplied by 3), not preserved overall. - Parallelism: Preserved by both transformations, so preserved overall. - Orientation: Preserved by dilation, but flipped by reflection, so NOT preserved overall. The answer is: angle measures and parallelism.