Scale Factor and Dilations

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Answer Key & Explanations

Scale Factor and Dilations · Grade 10 · Worksheet 2

1. A rectangular garden has dimensions 12 feet by 18 feet. If the garden undergoes a dilation with scale factor k, and the area of the dilated garden is 864 square feet, what is the value of k? Answer: 2 Solution: Find the original area of the garden. Original area = 12 × 18 = 216 square feet. When a figure is dilated by scale factor k, the area changes by a factor of k².
   Full step-by-step solution

   Step 1: Find the original area of the garden. Original area = 12 × 18 = 216 square feet. Step 2: When a figure is dilated by scale factor k, the area changes by a factor of k². Step 3: Set up the equation: original area × k² = dilated area. So 216 × k² = 864. Step 4: Solve for k²: k² = 864 ÷ 216 = 4. Step 5: Solve for k: k = sqrt(4) = 2. The scale factor is 2.

2. Dilate point (11, 13) by scale factor 3 from center (3, 5). Find the new coordinates. Answer: (27, 29) Solution: Find the vector from the center (3, 5) to the point (11, 13). Horizontal distance: 11 - 3 = 8. Vertical distance: 13 - 5 = 8.
   Full step-by-step solution

   Step 1: Find the vector from the center (3, 5) to the point (11, 13). Horizontal distance: 11 - 3 = 8. Vertical distance: 13 - 5 = 8. So the vector is (8, 8). Step 2: Multiply the vector by the scale factor 3: (8 × 3, 8 × 3) = (24, 24). Step 3: Add this scaled vector to the center coordinates: (3 + 24, 5 + 24) = (27, 29). The answer is (27, 29).

3. Emma dilates point (7, 11) by scale factor 3 from the origin. Find the new coordinates. Answer: (21, 33) Solution: The original point is (7, 11) and the scale factor is 3. To dilate from the origin, multiply each coordinate by the scale factor.
   Full step-by-step solution

   Step 1: The original point is (7, 11) and the scale factor is 3. Step 2: To dilate from the origin, multiply each coordinate by the scale factor. Step 3: New x-coordinate: 7 × 3 = 21 Step 4: New y-coordinate: 11 × 3 = 33 Step 5: The new coordinates are (21, 33). The answer is (21, 33).

4. Sophia dilates point (9, 7) by scale factor 4 from the origin. Find the new coordinates. Answer: (36, 28) Solution: The original point is (9, 7) and the scale factor is 4.
   Full step-by-step solution

   Step 1: The original point is (9, 7) and the scale factor is 4. Step 2: Multiply the x-coordinate by the scale factor: 9 × 4 = 36 Step 3: Multiply the y-coordinate by the scale factor: 7 × 4 = 28 Step 4: The new coordinates are (36, 28) The answer is (36, 28).

5. Dilate point (8, 10) by scale factor 2 from origin Answer: (16, 20) Solution: The dilation formula from the origin is (x', y') = (k × x, k × y), where k is the scale factor.
   Full step-by-step solution

   Step 1: The dilation formula from the origin is (x', y') = (k × x, k × y), where k is the scale factor. Step 2: Multiply the x-coordinate by the scale factor: 8 × 2 = 16 Step 3: Multiply the y-coordinate by the scale factor: 10 × 2 = 20 Step 4: The dilated point is (16, 20) The answer is (16, 20).

6. A cartographer is creating a detailed map of a national park. The original blueprint of the park's main trail system is drawn on a coordinate grid. The triangular region formed by points A(2, 4), B(6, 8), and C(10, 2) represents a forested area. To fit this region onto the final map, the cartographer applies a dilation centered at the origin with a scale factor of 2.5. What are the coordinates of the vertices of the dilated triangular region? Answer: A'(5, 10), B'(15, 20), C'(25, 5) Solution: We have a dilation centered at the origin with a scale factor of 2.5. Multiply each coordinate of each point by the scale factor.
   Full step-by-step solution

   Let's go step by step. We have a dilation centered at the origin with a scale factor of 2.5. **Rule for dilation centered at origin:** Multiply each coordinate of each point by the scale factor. --- **Step 1: Apply dilation to point A(2, 4)** x' = 2 × 2.5 = 5 y' = 4 × 2.5 = 10 So A' = (5, 10) --- **Step 2: Apply dilation to point B(6, 8)** x' = 6 × 2.5 = 15 y' = 8 × 2.5 = 20 So B' = (15, 20) --- **Step 3: Apply dilation to point C(10, 2)** x' = 10 × 2.5 = 25 y' = 2 × 2.5 = 5 So C' = (25, 5) --- **Final Answer:** A'(5, 10), B'(15, 20), C'(25, 5)

7. Noah is an engineer designing a triangular support bracket for a bridge. On his coordinate grid, the bracket has vertices at A(6, 16), B(26, 36), and C(46, 6). To test a smaller prototype for a different load condition, he applies a dilation centered at the origin with a scale factor of 1/2. What are the coordinates of vertex C' after this dilation? Answer: (23, 3) Solution: Identify the original coordinates of vertex C: (46, 6). The dilation is centered at the origin with scale factor k = 1/2. Multiply the x-coordinate: x' = (1/2) * 46 = 23.
   Full step-by-step solution

   Step 1: Identify the original coordinates of vertex C: (46, 6). Step 2: The dilation is centered at the origin with scale factor k = 1/2. Step 3: Apply the dilation formula: (x', y') = (k * x, k * y). Step 4: Multiply the x-coordinate: x' = (1/2) * 46 = 23. Step 5: Multiply the y-coordinate: y' = (1/2) * 6 = 3. Step 6: The coordinates of vertex C' after dilation are (23, 3). The answer is (23, 3).

8. Dilate point (6, 11) by scale factor 2 from center (1, 1) Answer: (11, 21) Solution: Find the vector from the center (1, 1) to the point (6, 11): (6 - 1, 11 - 1) = (5, 10) Multiply this vector by the scale factor 2: (5 × 2, 10 × 2) = (10, 20) Add this result to the center coordinates: (1 + 10, 1 + 20) = (11, 21) The answer is (11, 21).
   Full step-by-step solution

   Step 1: Find the vector from the center (1, 1) to the point (6, 11): (6 - 1, 11 - 1) = (5, 10) Step 2: Multiply this vector by the scale factor 2: (5 × 2, 10 × 2) = (10, 20) Step 3: Add this result to the center coordinates: (1 + 10, 1 + 20) = (11, 21) The answer is (11, 21).