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. 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

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.