Similarity Concepts

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Answer Key & Explanations

Similarity Concepts · Grade 8 · Worksheet 3

1. (4.2 × 10^5) ÷ (7 × 10^2) = ? Answer: 600 Solution: Separate the problem into two parts: (4.2 ÷ 7) and (10^5 ÷ 10^2). Calculate 4.2 ÷ 7 = 0.6. Calculate 10^5 ÷ 10^2 = 10^(5-2) = 10^3.
   Full step-by-step solution

   Step 1: Separate the problem into two parts: (4.2 ÷ 7) and (10^5 ÷ 10^2). Step 2: Calculate 4.2 ÷ 7 = 0.6. Step 3: Calculate 10^5 ÷ 10^2 = 10^(5-2) = 10^3. Step 4: Multiply the results: 0.6 × 10^3 = 0.6 × 1000 = 600. The answer is 600.

2. Liam is designing a triangular logo for his school's robotics team. The original triangle has side lengths of 6 cm, 8 cm, and 10 cm. He wants to create a similar but larger version where the shortest side becomes 15 cm. What will be the perimeter of the enlarged logo? Answer: 60 cm Solution: Identify the shortest side in the original triangle. The original triangle has sides 6 cm, 8 cm, and 10 cm. The shortest side is 6 cm.
   Full step-by-step solution

   Step 1: Identify the shortest side in the original triangle. The original triangle has sides 6 cm, 8 cm, and 10 cm. The shortest side is 6 cm. Step 2: Find the scale factor for the enlargement. The enlarged logo's shortest side is 15 cm. Scale factor = (New shortest side) / (Original shortest side) = 15 / 6 = 5/2 = 2.5. Step 3: Find the lengths of the other sides in the enlarged triangle. Multiply each original side by the scale factor 5/2. - Second side: 8 * (5/2) = 40/2 = 20 cm. - Third side: 10 * (5/2) = 50/2 = 25 cm. Step 4: Calculate the perimeter of the enlarged triangle. Perimeter = Sum of all sides = 15 + 20 + 25 = 60 cm. Step 5: Conclusion. The perimeter of the enlarged logo is 60 cm.

3. (2.4 × 10³) × (3.5 × 10⁻²) = ? Answer: 84 Solution: Multiply the coefficients: 2.4 × 3.5 = 8.4 Add the exponents: 3 + (-2) = 1 Combine the results: 8.4 × 10¹ Convert to standard form: 8.4 × 10 = 84 The answer is 84.
   Full step-by-step solution

   Step 1: Multiply the coefficients: 2.4 × 3.5 = 8.4 Step 2: Add the exponents: 3 + (-2) = 1 Step 3: Combine the results: 8.4 × 10¹ Step 4: Convert to standard form: 8.4 × 10 = 84 The answer is 84.

4. Liam is designing a logo for his robotics team. He starts with a triangle that has vertices at (0,0), (4,0), and (2,3). He wants to create a larger, similar version of this triangle by applying a dilation with a scale factor of 2.5, centered at the origin. After the dilation, what are the coordinates of the new triangle's vertices? Answer: (0,0), (10,0), (5,7.5) Solution: A dilation centered at the origin with scale factor \( k \) multiplies each coordinate by \( k \). Here, scale factor \( k = 2.5 \).
   Full step-by-step solution

   Let's go step by step. --- **Step 1: Understand the dilation rule** A dilation centered at the origin with scale factor \( k \) multiplies each coordinate by \( k \). Here, scale factor \( k = 2.5 \). So for any point \((x, y)\), the new point is: \[ (x', y') = (2.5 \times x, 2.5 \times y) \] --- **Step 2: Apply to each vertex** **Vertex A: (0, 0)** \[ x' = 2.5 \times 0 = 0 \] \[ y' = 2.5 \times 0 = 0 \] New coordinates: (0, 0) --- **Vertex B: (4, 0)** \[ x' = 2.5 \times 4 = 10 \] \[ y' = 2.5 \times 0 = 0 \] New coordinates: (10, 0) --- **Vertex C: (2, 3)** \[ x' = 2.5 \times 2 = 5 \] \[ y' = 2.5 \times 3 = 7.5 \] New coordinates: (5, 7.5) --- **Step 3: Final answer** The new triangle's vertices are: (0, 0), (10, 0), (5, 7.5)

5. Liam is designing a scale model of a new playground for his school project. The actual playground will be a rectangular area measuring 120 feet by 90 feet. Liam's model uses a scale where 1 inch represents 15 feet. What are the dimensions, in inches, of the rectangular base for his model? Answer: 8 inches by 6 inches Solution: The scale is: 1 inch in the model represents 15 feet in real life. Real length (feet) ÷ 15 = Model length (inches). Actual length = 120 feet.
   Full step-by-step solution

   Let's go step-by-step. --- **Step 1: Understand the scale** The scale is: 1 inch in the model represents 15 feet in real life. This means: Real length (feet) ÷ 15 = Model length (inches). --- **Step 2: Find the model length** Actual length = 120 feet. Model length = 120 / 15 inches. 120 ÷ 15 = 8 inches. --- **Step 3: Find the model width** Actual width = 90 feet. Model width = 90 / 15 inches. 90 ÷ 15 = 6 inches. --- **Step 4: State the answer** The dimensions of the rectangular base for the model are: 8 inches by 6 inches. --- **Final answer:** 8 inches by 6 inches

6. A triangle is drawn on a coordinate plane with vertices at A(2, 3), B(6, 3), and C(2, 7). This triangle is first reflected across the y-axis, then dilated by a scale factor of 2 with the origin as the center of dilation. What are the coordinates of the final image of vertex C after both transformations? Answer: (-4, 14) Solution: Start with the original coordinates of vertex C: (2, 7) Apply reflection across the y-axis. Reflection across the y-axis changes the sign of the x-coordinate while keeping the y-coordinate the same.
   Full step-by-step solution

   Step 1: Start with the original coordinates of vertex C: (2, 7) Step 2: Apply reflection across the y-axis. Reflection across the y-axis changes the sign of the x-coordinate while keeping the y-coordinate the same. So (2, 7) becomes (-2, 7). Step 3: Apply dilation with scale factor 2 and center at the origin. Dilation multiplies both coordinates by the scale factor. So (-2, 7) becomes (-2 × 2, 7 × 2) = (-4, 14). Step 4: The final coordinates after both transformations are (-4, 14).