Transformation Properties

lessonbunny.com

Answer Key & Explanations

Transformation Properties · Grade 8 · Worksheet 3

1. Liam is designing a rectangular garden with a length of 12 meters and a width of 8 meters. He wants to create a scale drawing of the garden using a scale factor of 1:50. What will be the perimeter of the garden in centimeters on his scale drawing? Answer: 80 Solution: The scale is 1:50, meaning 1 unit on the drawing represents 50 units in real life.
   Full step-by-step solution

   Let's go step-by-step. --- **Step 1: Understand the scale factor** The scale is 1:50, meaning 1 unit on the drawing represents 50 units in real life. --- **Step 2: Find the scaled length and width in meters** Real length = 12 m Real width = 8 m Scaled length = 12 m / 50 = 0.24 m Scaled width = 8 m / 50 = 0.16 m --- **Step 3: Convert scaled dimensions to centimeters** 1 m = 100 cm Scaled length in cm = 0.24 m × 100 = 24 cm Scaled width in cm = 0.16 m × 100 = 16 cm --- **Step 4: Find the perimeter of the scaled rectangle in cm** Perimeter formula for a rectangle: P = 2 × (length + width) P = 2 × (24 cm + 16 cm) P = 2 × (40 cm) P = 80 cm --- **Step 5: Conclusion** The perimeter of the garden on the scale drawing is **80 cm**. --- **Final answer:** 80

2. Mere is designing a triangular logo for her school's sports day. She draws a triangle with side lengths of 30 cm, 40 cm, and 50 cm, and angles of 90°, 53°, and 37°. She then reflects the triangle across a vertical line, and then dilates the reflected image by a scale factor of 3/5, using the origin as the center of dilation. After both transformations are applied, are the angles and side lengths of the final triangle preserved compared to the original? For each property (angles and side lengths), state whether it is preserved or not, and briefly explain why. Answer: Angles are preserved; side lengths are not preserved. Solution: Analyze the first transformation (reflection). Reflection is a rigid motion (isometry). So after reflection, the triangle has the same angles (90°, 53°, 37°) and the same side lengths (30 cm, 40 cm, 50 cm).
   Full step-by-step solution

   Step 1: Analyze the first transformation (reflection). Reflection is a rigid motion (isometry). It preserves both angles and side lengths. So after reflection, the triangle has the same angles (90°, 53°, 37°) and the same side lengths (30 cm, 40 cm, 50 cm). Step 2: Analyze the second transformation (dilation by scale factor 3/5). Dilation is a similarity transformation. It preserves angles, so the angles remain 90°, 53°, and 37°. However, dilation changes all side lengths by multiplying them by the scale factor. The new side lengths become: 30 × 3/5 = 18 cm, 40 × 3/5 = 24 cm, and 50 × 3/5 = 30 cm. Step 3: Conclusion. Angles are preserved through both transformations (reflection preserves angles, dilation preserves angles). Side lengths are preserved through reflection but NOT through dilation. Therefore, in the final triangle, angles are preserved, but side lengths are not preserved. The answer is: Angles are preserved; side lengths are not preserved.

3. (2.5 × 10³) × (4 × 10⁻²) ÷ (5 × 10²) = ? Answer: 0.2 Solution: Multiply the coefficients: 2.5 × 4 = 10 Multiply the powers of 10: 10³ × 10⁻² = 10¹ (since 3 + (-2) = 1) Now we have 10 × 10¹ ÷ (5 × 10²) Divide the coefficients: 10 ÷ 5 = 2 Divide the powers of 10: 10¹ ÷ 10² = 10⁻¹ (since 1 - 2 = -1) Combine the results: 2 × 10⁻¹ = 0.2 The answer is 0.2.
   Full step-by-step solution

   Step 1: Multiply the coefficients: 2.5 × 4 = 10 Step 2: Multiply the powers of 10: 10³ × 10⁻² = 10¹ (since 3 + (-2) = 1) Step 3: Now we have 10 × 10¹ ÷ (5 × 10²) Step 4: Divide the coefficients: 10 ÷ 5 = 2 Step 5: Divide the powers of 10: 10¹ ÷ 10² = 10⁻¹ (since 1 - 2 = -1) Step 6: Combine the results: 2 × 10⁻¹ = 0.2 The answer is 0.2.

4. Sophia drew a quadrilateral on a coordinate plane with vertices at A(4, 2), B(10, 2), C(8, 7), and D(6, 7). She then reflected the quadrilateral across the y-axis to create a new image. After that, she dilated the reflected image by a scale factor of 2 centered at the origin. Are the side lengths and angle measures preserved after both transformations? Explain your reasoning. Answer: Angle measures are preserved, but side lengths are not preserved. Solution: Analyze the first transformation - reflection across the y-axis. A reflection is a rigid motion (isometry), which preserves both distances (side lengths) and angle measures. Analyze the second transformation - dilation by a scale factor of 2 centered at the origin.
   Full step-by-step solution

   Step 1: Analyze the first transformation - reflection across the y-axis. A reflection is a rigid motion (isometry), which preserves both distances (side lengths) and angle measures. Step 2: Analyze the second transformation - dilation by a scale factor of 2 centered at the origin. A dilation is NOT a rigid motion; it changes distances by multiplying all side lengths by the scale factor (in this case, 2). However, dilations do preserve angle measures. Step 3: Combine the effects. After the reflection, side lengths are unchanged. After the dilation, all side lengths are multiplied by 2, so they are NOT preserved (they become twice as long). Angle measures are preserved through both transformations (reflection preserves them, and dilation preserves them). Step 4: Final conclusion: Angle measures are preserved after both transformations, but side lengths are not preserved because the dilation changes distances. The answer is: Angle measures are preserved, but side lengths are not preserved.

5. 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.