Transformation Properties

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Answer Key & Explanations

Transformation Properties · Grade 8 · Worksheet 2

1. A rectangular garden is transformed by a scale factor of 2.5. The original garden has a length of 8 meters and a width of 6 meters. After the transformation, what is the area of the new garden in square meters? Answer: 300 Solution: Find the area of the original garden. The original length is 8 meters and the original width is 6 meters. Area = length × width = 8 × 6 = 48 square meters.
   Full step-by-step solution

   Step 1: Find the area of the original garden. The original length is 8 meters and the original width is 6 meters. Area = length × width = 8 × 6 = 48 square meters. Step 2: Understand the effect of the scale factor on area. When a shape is scaled by a factor k, the area changes by a factor of k². Here, the scale factor k = 2.5, so the area scale factor is (2.5)². Step 3: Calculate the area scale factor. (2.5)² = 2.5 × 2.5 = 6.25. Step 4: Find the area of the new garden. New area = original area × area scale factor = 48 × 6.25. Step 5: Perform the multiplication. 48 × 6.25 = 48 × (6 + 0.25) = (48 × 6) + (48 × 0.25) 48 × 6 = 288 48 × 0.25 = 48 × (1/4) = 48 ÷ 4 = 12 So, 288 + 12 = 300. Step 6: State the final answer. The area of the new garden is 300 square meters.

2. A triangle has vertices at A(1, 1), B(6, 1), and C(1, 6). After a dilation with center at the origin and scale factor 2, followed by a reflection over the y-axis, which properties are preserved: angle measures, side lengths, and/or parallelism? Answer: Angle measures and parallelism are preserved; side lengths are not preserved. Solution: Apply dilation with scale factor 2. New coordinates: A'(2,2), B'(12,2), C'(2,12). Side lengths double: AB was 5, now 10; AC was 5, now 10; BC was sqrt(50) ≈ 7.07, now sqrt(200) ≈ 14.14.
   Full step-by-step solution

   Step 1: Apply dilation with scale factor 2. New coordinates: A'(2,2), B'(12,2), C'(2,12). Side lengths double: AB was 5, now 10; AC was 5, now 10; BC was sqrt(50) ≈ 7.07, now sqrt(200) ≈ 14.14. Angles remain the same because dilation is a similarity transformation. Parallelism is preserved because lines remain straight and parallel lines remain parallel. Step 2: Apply reflection over y-axis. New coordinates: A''(-2,2), B''(-12,2), C''(-2,12). Reflection is an isometry, so side lengths remain the same as after dilation, and angles remain unchanged. Parallelism is also preserved. Step 3: Compare original and final: Original side lengths (5, 5, sqrt(50)) changed to (10, 10, sqrt(200)) — not preserved. Original angles (90°, 45°, 45°) remain 90°, 45°, 45° — preserved. Original parallel sides (none in this triangle) would have stayed parallel if present. The answer is: Angle measures and parallelism are preserved; side lengths are not preserved.

3. Liam is designing a rectangular garden with a length of 12 feet and a width of 8 feet. He wants to create a scale drawing of the garden where 1 inch on the drawing represents 4 feet in the actual garden. What will be the perimeter, in inches, of the garden on his scale drawing? Answer: 10 Solution: First, find the actual perimeter of the garden in feet. The actual length is 12 feet and the actual width is 8 feet.
   Full step-by-step solution

   First, find the actual perimeter of the garden in feet. The actual length is 12 feet and the actual width is 8 feet. Perimeter formula for a rectangle is: Perimeter = 2 × (length + width) So, Actual perimeter = 2 × (12 + 8) Actual perimeter = 2 × 20 Actual perimeter = 40 feet. Now, the scale is 1 inch on the drawing = 4 feet in reality. We need the perimeter in inches on the drawing. Since perimeter is a length, we can convert the actual perimeter to drawing inches using the scale: Scale factor for length (and perimeter) is: 1 inch / 4 feet So, Perimeter on drawing = Actual perimeter × (1 inch / 4 feet) Perimeter on drawing = 40 feet × (1 inch / 4 feet) Perimeter on drawing = 40 / 4 inches Perimeter on drawing = 10 inches. Thus, the perimeter on the scale drawing is 10 inches.

4. Noah is designing a triangular park for his town. The park has vertices at points A(7, 11), B(19, 11), and C(7, 23) on a coordinate grid where each unit represents 1 meter. He wants to apply a dilation centered at the origin with a scale factor of 3 to enlarge the park, and then reflect the enlarged park across the y-axis. After both transformations, will the angles of the triangular park remain the same as the original? Will the side lengths be preserved? Explain your reasoning, and then calculate the perimeter of the final triangle after both transformations. Answer: Angles are preserved; side lengths are not preserved (they are multiplied by 3); perimeter of final triangle = 144 meters Solution: Identify the original triangle's side lengths. Side AB: from (7,11) to (19,11) is horizontal, length = 19 - 7 = 12 meters. Side AC: from (7,11) to (7,23) is vertical, length = 23 - 11 = 12 meters.
   Full step-by-step solution

   Step 1: Identify the original triangle's side lengths. Side AB: from (7,11) to (19,11) is horizontal, length = 19 - 7 = 12 meters. Side AC: from (7,11) to (7,23) is vertical, length = 23 - 11 = 12 meters. Side BC: from (19,11) to (7,23). Use distance formula: sqrt((19-7)^2 + (11-23)^2) = sqrt(12^2 + (-12)^2) = sqrt(144 + 144) = sqrt(288) = sqrt(144*2) = 12*sqrt(2) ≈ 16.97 meters. Original perimeter = 12 + 12 + 12*sqrt(2) = 24 + 12*sqrt(2) meters. Step 2: Apply dilation with scale factor 3 centered at origin. Dilation multiplies all distances by 3. Angles are preserved under dilation. New side lengths: AB = 36 m, AC = 36 m, BC = 36*sqrt(2) m. Perimeter after dilation = 36 + 36 + 36*sqrt(2) = 72 + 36*sqrt(2) meters. Step 3: Apply reflection across the y-axis. Reflection preserves both distances and angles. Side lengths remain: AB = 36 m, AC = 36 m, BC = 36*sqrt(2) m. Angles remain the same as after dilation (which were same as original). Step 4: Answer the questions. Angles: Yes, angles are preserved through both transformations. Side lengths: No, side lengths are not preserved because dilation changed them by a factor of 3. Perimeter of final triangle = 72 + 36*sqrt(2) meters. To get a decimal: 36 * 1.4142 = 50.91, so 72 + 50.91 = 122.91 meters. For the exact answer: perimeter = 72 + 36*sqrt(2) meters. The answer is: Angles are preserved; side lengths are not preserved; perimeter of final triangle = 72 + 36*sqrt(2) meters (approximately 122.9 meters).