Similarity Concepts

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Similarity Concepts · Grade 8 · Worksheet 2

1. Liam is designing a scale model of a new park for his city planning project. The actual park will be a rectangular area measuring 240 meters by 180 meters. Liam's model uses a scale where 1 centimeter represents 15 meters. What are the dimensions of Liam's model in centimeters? Answer: 16 cm by 12 cm Solution: The scale is: 1 cm in the model represents 15 meters in reality. 1 cm → 15 m So to convert actual meters to model centimeters, we divide by 15.
   Full step-by-step solution

   Let's go step by step. --- **Step 1: Understand the scale** The scale is: 1 cm in the model represents 15 meters in reality. That means: 1 cm → 15 m So to convert actual meters to model centimeters, we divide by 15. --- **Step 2: Convert the first dimension (240 meters)** Actual length = 240 m Model length = 240 / 15 cm 240 ÷ 15 = 16 So model length = 16 cm. --- **Step 3: Convert the second dimension (180 meters)** Actual width = 180 m Model width = 180 / 15 cm 180 ÷ 15 = 12 So model width = 12 cm. --- **Step 4: Conclusion** The model dimensions are: 16 cm by 12 cm

2. A right triangle has vertices at coordinates A(0,0), B(6,0), and C(0,8). A similar triangle is created by applying a dilation with a scale factor of 1.5, followed by a translation 2 units to the right and 3 units up. What are the coordinates of the image of point C after both transformations? Answer: (2,15) Solution: A(0,0), B(6,0), C(0,8) So C = (0, 8) Apply dilation with scale factor 1.5 A dilation centered at the origin (0,0) multiplies both coordinates by the scale factor.
   Full step-by-step solution

   Let's go step-by-step. --- **Step 1: Identify original coordinates of point C** From the problem: A(0,0), B(6,0), C(0,8) So C = (0, 8) --- **Step 2: Apply dilation with scale factor 1.5** A dilation centered at the origin (0,0) multiplies both coordinates by the scale factor. New coordinates after dilation: x' = 0 × 1.5 = 0 y' = 8 × 1.5 = 12 So after dilation: C' = (0, 12) --- **Step 3: Apply translation 2 units right and 3 units up** Translation rule: (x, y) → (x + 2, y + 3) Apply to C' = (0, 12): x'' = 0 + 2 = 2 y'' = 12 + 3 = 15 So after translation: C'' = (2, 15) --- **Step 4: Final answer** The image of point C after both transformations is (2, 15).

3. A rectangular garden is drawn on a coordinate plane with vertices at (2, 1), (8, 1), (8, 5), and (2, 5). The garden is dilated by a scale factor of 2.5 with the origin as the center of dilation. What are the coordinates of the vertices of the dilated garden? Answer: (5, 2.5), (20, 2.5), (20, 12.5), (5, 12.5) Solution: A = (2, 1) B = (8, 1) C = (8, 5) D = (2, 5) The center of dilation is the origin (0, 0). The scale factor is 2.5.
   Full step-by-step solution

   Let's go step-by-step. --- **Step 1: Understand the problem** We have a rectangle with vertices: A = (2, 1) B = (8, 1) C = (8, 5) D = (2, 5) The center of dilation is the origin (0, 0). The scale factor is 2.5. Dilation rule with center at origin: (x, y) → (2.5 * x, 2.5 * y) --- **Step 2: Apply dilation to each vertex** For A = (2, 1): New x = 2.5 * 2 = 5 New y = 2.5 * 1 = 2.5 A' = (5, 2.5) For B = (8, 1): New x = 2.5 * 8 = 20 New y = 2.5 * 1 = 2.5 B' = (20, 2.5) For C = (8, 5): New x = 2.5 * 8 = 20 New y = 2.5 * 5 = 12.5 C' = (20, 12.5) For D = (2, 5): New x = 2.5 * 2 = 5 New y = 2.5 * 5 = 12.5 D' = (5, 12.5) --- **Step 3: Write the final answer in order** The dilated rectangle's vertices are: (5, 2.5), (20, 2.5), (20, 12.5), (5, 12.5) --- **Step 4: Check** Original rectangle: width = 8 - 2 = 6, height = 5 - 1 = 4. After dilation: width = 20 - 5 = 15, height = 12.5 - 2.5 = 10. Scale factor check: 15/6 = 2.5, 10/4 = 2.5. Correct. --- **Final Answer:** (5, 2.5), (20, 2.5), (20, 12.5), (5, 12.5)

4. A triangle has vertices at A(2, 3), B(6, 3), and C(2, 7). It 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) Reflect across the y-axis - this changes the sign of the x-coordinate while keeping the y-coordinate the same: (-2, 7) Apply dilation with scale factor 2 from the origin - multiply both coordinates by 2: (-2 × 2, 7 × 2) = (-4, 14) The final…
   Full step-by-step solution

   Step 1: Start with the original coordinates of vertex C: (2, 7) Step 2: Reflect across the y-axis - this changes the sign of the x-coordinate while keeping the y-coordinate the same: (-2, 7) Step 3: Apply dilation with scale factor 2 from the origin - multiply both coordinates by 2: (-2 × 2, 7 × 2) = (-4, 14) Step 4: The final coordinates after both transformations are (-4, 14)

5. (3.6 × 10⁸) ÷ (1.2 × 10³) × (4 × 10²) = ? Answer: 120000000 Solution: Divide the coefficients: 3.6 ÷ 1.2 = 3 Subtract the exponents for division: 10^(8-3) = 10^5 Multiply the intermediate result by the next term: 3 × 10^5 × (4 × 10^2) Multiply the coefficients: 3 × 4 = 12 Add the exponents: 10^(5+2) = 10^7 Combine the results: 12 × 10^7 Convert to standard form:…
   Full step-by-step solution

   Step 1: Divide the coefficients: 3.6 ÷ 1.2 = 3 Step 2: Subtract the exponents for division: 10^(8-3) = 10^5 Step 3: Multiply the intermediate result by the next term: 3 × 10^5 × (4 × 10^2) Step 4: Multiply the coefficients: 3 × 4 = 12 Step 5: Add the exponents: 10^(5+2) = 10^7 Step 6: Combine the results: 12 × 10^7 Step 7: Convert to standard form: 12 × 10,000,000 = 120,000,000 The answer is 120000000.

6. A triangle has vertices at A(1, 1), B(4, 1), and C(1, 5). It is dilated by a scale factor of 3 with the origin as the center of dilation. What are the coordinates of the image point C'? Answer: (3, 15) Solution: When a point (x, y) is dilated about the origin by a scale factor k, the new coordinates are (k*x, k*y). Here, the scale factor is 3, so the rule is: (x, y) → (3x, 3y). Identify the coordinates of point C.
   Full step-by-step solution

   Step 1: Understand the dilation rule. When a point (x, y) is dilated about the origin by a scale factor k, the new coordinates are (k*x, k*y). Here, the scale factor is 3, so the rule is: (x, y) → (3x, 3y). Step 2: Identify the coordinates of point C. From the problem, C is at (1, 5). Step 3: Apply the dilation rule to point C. Multiply the x-coordinate by 3: 3 * 1 = 3. Multiply the y-coordinate by 3: 3 * 5 = 15. Step 4: Write the new coordinates. C' = (3, 15). Step 5: Check the answer. The problem asks for the image of C after dilation, and we have computed it as (3, 15). This matches the correct answer.