Similar Figures

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Answer Key & Explanations

Similar Figures · Grade 7 · Worksheet 3

1. Liam is designing a scale model of a new playground for his town. The actual playground will be a rectangular area measuring 120 feet by 90 feet. Liam's blueprint uses a scale where 1 inch represents 15 feet. What will be the perimeter of the playground on his blueprint, in inches? Answer: 28 Solution: The scale is 1 inch = 15 feet. This means every 15 feet in real life is shown as 1 inch on the blueprint.
   Full step-by-step solution

   Step 1: Understand the scale The scale is 1 inch = 15 feet. This means every 15 feet in real life is shown as 1 inch on the blueprint. Step 2: Find the blueprint dimensions Actual length = 120 feet Blueprint length = 120 / 15 = 8 inches Actual width = 90 feet Blueprint width = 90 / 15 = 6 inches Step 3: Find the perimeter on the blueprint Perimeter formula for a rectangle: P = 2 × (length + width) Blueprint length = 8 inches, Blueprint width = 6 inches P = 2 × (8 + 6) P = 2 × 14 P = 28 inches Step 4: Conclusion The perimeter of the playground on the blueprint is 28 inches.

2. Aroha is designing a scale model of a rectangular garden. The actual garden is 12 m long and 9 m wide. In her model, the length is 16 cm. What is the width of the model in centimeters? Answer: 12 Solution: Write the scale factor as a ratio of model length to actual length. Model length = 16 cm, actual length = 12 m = 1200 cm. Let w be the model width in cm.
   Full step-by-step solution

   Step 1: Write the scale factor as a ratio of model length to actual length. Model length = 16 cm, actual length = 12 m = 1200 cm. So the scale factor is 16/1200 = 1/75. Step 2: Let w be the model width in cm. The actual width is 9 m = 900 cm. Step 3: Set up the proportion: w/900 = 16/1200. Step 4: Simplify the right side: 16/1200 = 1/75. Step 5: Solve for w: w = 900 × (1/75) = 900/75 = 12. Step 6: The width of the model is 12 cm. The answer is 12.

3. Emma is creating a scale drawing of her school's rectangular library for a history project. The actual library measures 45 meters in length and 30 meters in width. If Emma's drawing has a length of 18 centimeters, what is the width of her drawing in centimeters? Answer: 12 Solution: Identify the scale factor using the length dimension. Actual length = 45 meters Drawing length = 18 centimeters Scale factor = drawing length / actual length = 18 cm / 45 m Apply the same scale factor to find the drawing width.
   Full step-by-step solution

   Step 1: Identify the scale factor using the length dimension. Actual length = 45 meters Drawing length = 18 centimeters Scale factor = drawing length / actual length = 18 cm / 45 m Step 2: Apply the same scale factor to find the drawing width. Actual width = 30 meters Drawing width = actual width × scale factor = 30 m × (18 cm / 45 m) Step 3: Simplify the calculation. 30 × (18/45) = 30 × (2/5) = 60/5 = 12 Step 4: The width of Emma's drawing is 12 centimeters.

4. Two similar rectangles have corresponding sides in the ratio 5:12. If the perimeter of the smaller rectangle is 40 cm, what is the perimeter of the larger rectangle? Answer: 96 Solution: The ratio of corresponding sides is 5:12, so the scale factor from the smaller to the larger is 12/5. For similar figures, the ratio of perimeters equals the ratio of corresponding side lengths.
   Full step-by-step solution

   Step 1: The ratio of corresponding sides is 5:12, so the scale factor from the smaller to the larger is 12/5. Step 2: For similar figures, the ratio of perimeters equals the ratio of corresponding side lengths. Step 3: Let P be the perimeter of the larger rectangle. Then 40/P = 5/12. Step 4: Cross multiply: 40 × 12 = 5 × P Step 5: 480 = 5P Step 6: Divide both sides by 5: P = 480 ÷ 5 = 96 The perimeter of the larger rectangle is 96 cm.