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Scale Factor and Dilations

Grade 10 · Mathematics · Worksheet 2

  1. Aroha dilates point (3, 7) by scale factor 5 from center (1, 1). Find the new coordinates. Answer: ______________
  2. A graphic designer is creating a logo that will be printed on business cards and large banners. The original logo design is a triangle with vertices at (0,0), (4,0), and (2,3). For the banner, she applies a dilation centered at the origin with a scale factor of 2.5. What are the coordinates of the dilated triangle's vertices? Answer: ______________
  3. Dilate point (7,12) by scale factor 2 from origin = ? Answer: ______________
  4. A trapezoid is drawn on a coordinate plane with vertices at A(-7, 2), B(-2, 2), C(2, -7), and D(-7, -7). Charlotte dilates this trapezoid from the origin using a scale factor of 2/7. What are the coordinates of the dilated trapezoid's vertices? Express your answer as four ordered pairs: A', B', C', D'. Answer: ______________
  5. A cartographer is creating a detailed map of a national park. The original blueprint of a ranger station is drawn on a coordinate grid with vertices at A(2, 3), B(6, 3), C(6, 7), and D(2, 7). For the final map, she applies a dilation centered at the origin with a scale factor of 2.5. What are the coordinates of vertex C' after this dilation? Answer: ______________
  6. Dilate point (6, 11) by scale factor 4 from the origin Answer: ______________
  7. Olivia dilates point (4, 10) by scale factor 5 from the origin. Find the new coordinates. Answer: ______________
  8. Isabella dilates point (8, 12) by scale factor 5/4 from the origin. Find the new coordinates. Answer: ______________
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Answer Key & Explanations

Scale Factor and Dilations · Grade 10 · Worksheet 2

  1. Aroha dilates point (3, 7) by scale factor 5 from center (1, 1). Find the new coordinates. Answer: (11, 31) Solution: Find the horizontal distance from center (1, 1) to point (3, 7): 3 - 1 = 2 Find the vertical distance from center (1, 1) to point (3, 7): 7 - 1 = 6 Multiply horizontal distance by scale factor 5: 2 × 5 = 10 Multiply vertical distance by scale factor 5: 6 × 5 = 30 Add scaled horizontal distance…
    Full step-by-step solution

    Step 1: Find the horizontal distance from center (1, 1) to point (3, 7): 3 - 1 = 2 Step 2: Find the vertical distance from center (1, 1) to point (3, 7): 7 - 1 = 6 Step 3: Multiply horizontal distance by scale factor 5: 2 × 5 = 10 Step 4: Multiply vertical distance by scale factor 5: 6 × 5 = 30 Step 5: Add scaled horizontal distance to center's x-coordinate: 1 + 10 = 11 Step 6: Add scaled vertical distance to center's y-coordinate: 1 + 30 = 31 Step 7: The new coordinates are (11, 31) The answer is (11, 31).

  2. A graphic designer is creating a logo that will be printed on business cards and large banners. The original logo design is a triangle with vertices at (0,0), (4,0), and (2,3). For the banner, she applies a dilation centered at the origin with a scale factor of 2.5. What are the coordinates of the dilated triangle's vertices? Answer: (0,0), (10,0), (5,7.5) Solution: 1. This means we multiply each coordinate of every vertex by 2.5. 2.
    Full step-by-step solution

    Step-by-step solution: 1. Understand the dilation: The dilation is centered at the origin (0,0) with a scale factor of 2.5. This means we multiply each coordinate of every vertex by 2.5. 2. Apply the dilation to each vertex: First vertex: (0,0) Multiply both coordinates by 2.5: x = 0 × 2.5 = 0 y = 0 × 2.5 = 0 Result: (0,0) Second vertex: (4,0) Multiply both coordinates by 2.5: x = 4 × 2.5 = 10 y = 0 × 2.5 = 0 Result: (10,0) Third vertex: (2,3) Multiply both coordinates by 2.5: x = 2 × 2.5 = 5 y = 3 × 2.5 = 7.5 Result: (5,7.5) 3. Final answer: The dilated triangle's vertices are (0,0), (10,0), and (5,7.5). Note: The vertex at the origin (0,0) remains unchanged because it is the center of dilation. All other points move away from the origin by a factor of 2.5.

  3. Dilate point (7,12) by scale factor 2 from origin = ? Answer: (14,24) Solution: Identify the original point coordinates: (7,12) Identify the scale factor: 2 Multiply the x-coordinate by the scale factor: 7 × 2 = 14 Multiply the y-coordinate by the scale factor: 12 × 2 = 24 Write the new coordinates as an ordered pair: (14,24) The answer is (14,24).
    Full step-by-step solution

    Step 1: Identify the original point coordinates: (7,12) Step 2: Identify the scale factor: 2 Step 3: Multiply the x-coordinate by the scale factor: 7 × 2 = 14 Step 4: Multiply the y-coordinate by the scale factor: 12 × 2 = 24 Step 5: Write the new coordinates as an ordered pair: (14,24) The answer is (14,24).

  4. A trapezoid is drawn on a coordinate plane with vertices at A(-7, 2), B(-2, 2), C(2, -7), and D(-7, -7). Charlotte dilates this trapezoid from the origin using a scale factor of 2/7. What are the coordinates of the dilated trapezoid's vertices? Express your answer as four ordered pairs: A', B', C', D'. Answer: (-2, 4/7), (-4/7, 4/7), (4/7, -2), (-2, -2) Solution: The original vertices are A(-7, 2), B(-2, 2), C(2, -7), D(-7, -7). Step 2: Dilation from the origin with scale factor k = 2/7 means each coordinate is multiplied by 2/7.
    Full step-by-step solution

    Step 1: The original vertices are A(-7, 2), B(-2, 2), C(2, -7), D(-7, -7). Step 2: Dilation from the origin with scale factor k = 2/7 means each coordinate is multiplied by 2/7. Step 3: Calculate A': (-7 * 2/7, 2 * 2/7) = (-2, 4/7). Step 4: Calculate B': (-2 * 2/7, 2 * 2/7) = (-4/7, 4/7). Step 5: Calculate C': (2 * 2/7, -7 * 2/7) = (4/7, -2). Step 6: Calculate D': (-7 * 2/7, -7 * 2/7) = (-2, -2). Step 7: The dilated trapezoid's vertices are A'(-2, 4/7), B'(-4/7, 4/7), C'(4/7, -2), D'(-2, -2).

  5. A cartographer is creating a detailed map of a national park. The original blueprint of a ranger station is drawn on a coordinate grid with vertices at A(2, 3), B(6, 3), C(6, 7), and D(2, 7). For the final map, she applies a dilation centered at the origin with a scale factor of 2.5. What are the coordinates of vertex C' after this dilation? Answer: (15, 17.5) Solution: A dilation centered at the origin multiplies each coordinate of a point by the scale factor. Scale factor given: 2.5 Identify the coordinates of vertex C before dilation.
    Full step-by-step solution

    Step 1: Understand the dilation transformation. A dilation centered at the origin multiplies each coordinate of a point by the scale factor. Scale factor given: 2.5 Step 2: Identify the coordinates of vertex C before dilation. From the problem: C(6, 7) Step 3: Apply the dilation to the x-coordinate. x' = scale factor × original x x' = 2.5 × 6 x' = 15 Step 4: Apply the dilation to the y-coordinate. y' = scale factor × original y y' = 2.5 × 7 y' = 17.5 Step 5: Write the new coordinates after dilation. C' = (15, 17.5) Final answer: (15, 17.5)

  6. Dilate point (6, 11) by scale factor 4 from the origin Answer: (24, 44) Solution: Identify the original point coordinates: (6, 11) Identify the scale factor: 4 Multiply the x-coordinate by the scale factor: 6 × 4 = 24 Multiply the y-coordinate by the scale factor: 11 × 4 = 44 Write the new coordinates: (24, 44) The answer is (24, 44).
    Full step-by-step solution

    Step 1: Identify the original point coordinates: (6, 11) Step 2: Identify the scale factor: 4 Step 3: Multiply the x-coordinate by the scale factor: 6 × 4 = 24 Step 4: Multiply the y-coordinate by the scale factor: 11 × 4 = 44 Step 5: Write the new coordinates: (24, 44) The answer is (24, 44).

  7. Olivia dilates point (4, 10) by scale factor 5 from the origin. Find the new coordinates. Answer: (20, 50) Solution: The original point is (4, 10) and the scale factor is 5.
    Full step-by-step solution

    Step 1: The original point is (4, 10) and the scale factor is 5. Step 2: Multiply the x-coordinate by the scale factor: 4 × 5 = 20 Step 3: Multiply the y-coordinate by the scale factor: 10 × 5 = 50 Step 4: The new coordinates are (20, 50) The answer is (20, 50).

  8. Isabella dilates point (8, 12) by scale factor 5/4 from the origin. Find the new coordinates. Answer: (10, 15) Solution: Identify the original point coordinates: (8, 12) Identify the scale factor: 5/4 Multiply the x-coordinate by the scale factor: 8 × 5/4 = 40/4 = 10 Multiply the y-coordinate by the scale factor: 12 × 5/4 = 60/4 = 15 Write the new coordinates: (10, 15) The answer is (10, 15).
    Full step-by-step solution

    Step 1: Identify the original point coordinates: (8, 12) Step 2: Identify the scale factor: 5/4 Step 3: Multiply the x-coordinate by the scale factor: 8 × 5/4 = 40/4 = 10 Step 4: Multiply the y-coordinate by the scale factor: 12 × 5/4 = 60/4 = 15 Step 5: Write the new coordinates: (10, 15) The answer is (10, 15).