Similar Figures

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Answer Key & Explanations

Similar Figures · Grade 7 · Worksheet 1

1. Tom is using a map with a scale of 1 inch = 56 miles. On the map, Tom measures the distance between two cities as 6 inches. What is the actual distance in miles between the two cities? Answer: 336 Solution: The scale means every 1 inch on the map represents 56 miles in real life. The map distance is 6 inches, so the actual distance is 6 × 56 = 336 miles. The actual distance between the cities is 336 miles.
   Full step-by-step solution

   Step 1: The scale means every 1 inch on the map represents 56 miles in real life. Step 2: The map distance is 6 inches, so the actual distance is 6 × 56 = 336 miles. Step 3: The actual distance between the cities is 336 miles.

2. Emma is creating a scale model of the Eiffel Tower for her history project. The actual Eiffel Tower is 330 meters tall. Emma's model uses a scale where 1 centimeter represents 22 meters. If she also needs to include the antenna on top, which adds 24 meters to the actual height, what will be the total height of her completed scale model in centimeters? Answer: 16.1 Solution: Find the total actual height including the antenna Actual Eiffel Tower height = 330 meters Antenna height = 24 meters Total actual height = 330 + 24 = 354 meters Scale: 1 cm represents 22 meters Model height = Total actual height ÷ Scale factor Model height = 354 ÷ 22 354 ÷ 22 = 16.0909...
   Full step-by-step solution

   Step 1: Find the total actual height including the antenna Actual Eiffel Tower height = 330 meters Antenna height = 24 meters Total actual height = 330 + 24 = 354 meters Step 2: Apply the scale ratio to find the model height Scale: 1 cm represents 22 meters Model height = Total actual height ÷ Scale factor Model height = 354 ÷ 22 Step 3: Calculate the division 354 ÷ 22 = 16.0909... Rounded to one decimal place = 16.1 Step 4: State the final answer The total height of Emma's completed scale model is 16.1 centimeters.

3. A scale drawing of a triangular park is made using a scale of 1 cm : 15 m. On the drawing, the triangle has a base of 6.4 cm and a height of 4.8 cm. What is the actual area of the park in square meters? Answer: 3456 Solution: Calculate the area of the triangle on the drawing. Area_drawing = (1/2) × base × height = (1/2) × 6.4 cm × 4.8 cm = (1/2) × 30.72 cm² = 15.36 cm² Determine the scale factor for area.
   Full step-by-step solution

   Step 1: Calculate the area of the triangle on the drawing. Area_drawing = (1/2) × base × height = (1/2) × 6.4 cm × 4.8 cm = (1/2) × 30.72 cm² = 15.36 cm² Step 2: Determine the scale factor for area. The linear scale is 1 cm : 15 m, which means 1 cm on the drawing represents 15 m in reality. For area, the scale factor is squared: (15)² = 225 Step 3: Calculate the actual area of the park. Actual area = Area_drawing × scale factor for area = 15.36 cm² × 225 = 3456 m² The answer is 3456 square meters.

4. Two similar rectangles have corresponding sides in the ratio 7:12. If the smaller rectangle has a perimeter of 42 cm, what is the perimeter of the larger rectangle? Answer: 72 Solution: The ratio of corresponding sides is 7:12, so the scale factor from the smaller to the larger rectangle is 12/7. For similar figures, the ratio of perimeters equals the ratio of corresponding sides.
   Full step-by-step solution

   Step 1: The ratio of corresponding sides is 7:12, so the scale factor from the smaller to the larger rectangle is 12/7. Step 2: For similar figures, the ratio of perimeters equals the ratio of corresponding sides. Step 3: Let P be the perimeter of the larger rectangle. Then 42/P = 7/12. Step 4: Cross multiply: 42 × 12 = 7 × P Step 5: 504 = 7P Step 6: Divide both sides by 7: P = 504 ÷ 7 = 72 The answer is 72.

5. Liam is designing a scale model of a new sports stadium for his architecture class. The actual stadium will be 180 meters long and 120 meters wide. Liam's model uses a scale where 1 centimeter represents 15 meters. What will be the perimeter of Liam's scale model in centimeters? Answer: 40 Solution: First, find the actual perimeter of the stadium. The actual length is 180 meters, and the actual width is 120 meters.
   Full step-by-step solution

   First, find the actual perimeter of the stadium. The actual length is 180 meters, and the actual width is 120 meters. Perimeter of a rectangle = 2 × (length + width) So actual perimeter = 2 × (180 + 120) = 2 × 300 = 600 meters. Now, the scale is 1 cm represents 15 meters. We need to convert the actual perimeter from meters to centimeters on the model. Since 1 cm on the model = 15 meters in reality, we divide the actual perimeter in meters by 15 to get the model perimeter in centimeters: 600 ÷ 15 = 40. Thus, the perimeter of Liam's scale model is 40 cm.

6. Maya is using a map with a scale of 1 inch = 18 miles. On the map, Maya measures the distance between two cities as 4 inches. What is the actual distance in miles between the two cities? Answer: 72 Solution: The scale means every 1 inch on the map represents 18 miles in real life. The map distance is 4 inches, so the actual distance is 4 × 18 = 72 miles. The actual distance between the cities is 72 miles.
   Full step-by-step solution

   Step 1: The scale means every 1 inch on the map represents 18 miles in real life. Step 2: The map distance is 4 inches, so the actual distance is 4 × 18 = 72 miles. Step 3: The actual distance between the cities is 72 miles.

7. Liam is creating a scale model of his school's basketball court for a project. The actual court measures 28 meters in length and 15 meters in width. If Liam's model has a length of 21 centimeters, what is the width of his model in centimeters? Answer: 11.25 Solution: We have the actual court dimensions: length = 28 m, width = 15 m. The model length = 21 cm. We need to find the model width in cm.
   Full step-by-step solution

   Step 1: Understand the problem We have the actual court dimensions: length = 28 m, width = 15 m. The model length = 21 cm. We need to find the model width in cm. Since it’s a scale model, the ratio of length to width must be the same in the model as in reality. Step 2: Set up the proportion Let the model width be \( w \) cm. The proportion is: (actual length) / (actual width) = (model length) / (model width) So: 28 / 15 = 21 / w Step 3: Solve for \( w \) Cross-multiply: 28 * w = 15 * 21 28 * w = 315 Divide both sides by 28: w = 315 / 28 Step 4: Simplify the fraction Divide numerator and denominator by 7: 315 ÷ 7 = 45 28 ÷ 7 = 4 So w = 45 / 4 Step 5: Convert to decimal 45 ÷ 4 = 11.25 Step 6: Conclusion The width of the model is 11.25 cm.

8. Emma is drawing a scale model of a triangular sail for a boat. The original sail has side lengths of 15 m, 21 m, and 27 m. She wants to create a similar smaller sail with a scale factor of 1/3. What will be the lengths of the corresponding sides in her scale drawing (in meters)? Answer: 5, 7, 9 Solution: Identify the scale factor, which is 1/3. Multiply each original side length by the scale factor.
   Full step-by-step solution

   Step 1: Identify the scale factor, which is 1/3. Step 2: Multiply each original side length by the scale factor. - First side: 15 m × 1/3 = 5 m - Second side: 21 m × 1/3 = 7 m - Third side: 27 m × 1/3 = 9 m Step 3: The corresponding sides of the smaller sail are 5 m, 7 m, and 9 m. The answer is 5, 7, 9.