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Scale Factor and Dilations

Grade 10 · Mathematics · Worksheet 1

  1. Hana is a landscape architect designing a triangular garden bed. The original design on her coordinate grid has vertices at A(12, 9), B(18, 21), and C(24, 3). To create a larger version for a public park, she applies a dilation centered at the origin with a scale factor of 4. What are the coordinates of vertex B' after this dilation? Answer: ______________
  2. Dilate point (9, 12) by scale factor 2/3 from center (3, 4) Answer: ______________
  3. 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: ______________
  4. Dilate point (8, 12) by scale factor 1.5 from origin Answer: ______________
  5. An architect is designing a new public library. The original blueprint shows a triangular reading nook with vertices at coordinates A(2, 1), B(6, 1), and C(4, 5). The architect decides to enlarge this space by applying a dilation with a scale factor of 2.5, centered at the origin (0, 0). What are the coordinates of vertex C after this dilation? Answer: ______________
  6. Liam is designing a logo for his robotics team. He creates a small prototype on graph paper with vertices at A(2, 3), B(6, 3), C(6, 7), and D(2, 7). For the final banner, he 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: ______________
  7. A right triangle is drawn on a coordinate plane with vertices at A(0,0), B(6,0), and C(0,8). The triangle is dilated from the point (2,1) with a scale factor of 3. What are the coordinates of the dilated triangle's vertices? Express your answer as three ordered pairs: A', B', C'. Answer: ______________
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Answer Key & Explanations

Scale Factor and Dilations · Grade 10 · Worksheet 1

  1. Hana is a landscape architect designing a triangular garden bed. The original design on her coordinate grid has vertices at A(12, 9), B(18, 21), and C(24, 3). To create a larger version for a public park, she applies a dilation centered at the origin with a scale factor of 4. What are the coordinates of vertex B' after this dilation? Answer: (72, 84) Solution: Identify the original coordinates of vertex B: (18, 21). Step 2: Since the dilation is centered at the origin with a scale factor of 4, multiply each coordinate by 4. Step 3: x-coordinate: 18 * 4 = 72.
    Full step-by-step solution

    Step 1: Identify the original coordinates of vertex B: (18, 21). Step 2: Since the dilation is centered at the origin with a scale factor of 4, multiply each coordinate by 4. Step 3: x-coordinate: 18 * 4 = 72. Step 4: y-coordinate: 21 * 4 = 84. Step 5: The coordinates of B' are (72, 84). The answer is (72, 84).

  2. Dilate point (9, 12) by scale factor 2/3 from center (3, 4) Answer: (7, 9.33) Solution: Identify the given values: point (9, 12), center (3, 4), scale factor = 2/3 Apply the dilation formula: x' = h + k(x - h) and y' = k + k(y - k) Calculate x-coordinate: x' = 3 + (2/3)(9 - 3) = 3 + (2/3)(6) = 3 + 4 = 7 Calculate y-coordinate: y' = 4 + (2/3)(12 - 4) = 4 + (2/3)(8) = 4 + 16/3 = 4 +…
    Full step-by-step solution

    Step 1: Identify the given values: point (9, 12), center (3, 4), scale factor = 2/3 Step 2: Apply the dilation formula: x' = h + k(x - h) and y' = k + k(y - k) Step 3: Calculate x-coordinate: x' = 3 + (2/3)(9 - 3) = 3 + (2/3)(6) = 3 + 4 = 7 Step 4: Calculate y-coordinate: y' = 4 + (2/3)(12 - 4) = 4 + (2/3)(8) = 4 + 16/3 = 4 + 5.33 = 9.33 Step 5: The dilated point is (7, 9.33) The answer is (7, 9.33).

  3. 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)

  4. Dilate point (8, 12) by scale factor 1.5 from origin Answer: (12, 18) Solution: Original point is (8, 12) Scale factor is 1.5 Multiply x-coordinate by scale factor: 8 × 1.5 = 12 Multiply y-coordinate by scale factor: 12 × 1.5 = 18 New coordinates are (12, 18) The answer is (12, 18).
    Full step-by-step solution

    Step 1: Original point is (8, 12) Step 2: Scale factor is 1.5 Step 3: Multiply x-coordinate by scale factor: 8 × 1.5 = 12 Step 4: Multiply y-coordinate by scale factor: 12 × 1.5 = 18 Step 5: New coordinates are (12, 18) The answer is (12, 18).

  5. An architect is designing a new public library. The original blueprint shows a triangular reading nook with vertices at coordinates A(2, 1), B(6, 1), and C(4, 5). The architect decides to enlarge this space by applying a dilation with a scale factor of 2.5, centered at the origin (0, 0). What are the coordinates of vertex C after this dilation? Answer: (10, 12.5) Solution: 1. Here, scale factor = 2.5. 2.
    Full step-by-step solution

    Step-by-step solution: 1. Understand the dilation: A dilation centered at the origin (0, 0) with scale factor k multiplies each coordinate of a point by k. Here, scale factor = 2.5. 2. Original coordinates of vertex C: C(4, 5) 3. Apply the dilation formula: For a point (x, y) dilated from the origin with scale factor k, the new coordinates are (k*x, k*y). So for C(4, 5) with k = 2.5: New x-coordinate = 2.5 * 4 New y-coordinate = 2.5 * 5 4. Calculate new x-coordinate: 2.5 * 4 = 10 5. Calculate new y-coordinate: 2.5 * 5 = 12.5 6. Final coordinates: C' = (10, 12.5) Thus, after dilation, vertex C is at (10, 12.5).

  6. Liam is designing a logo for his robotics team. He creates a small prototype on graph paper with vertices at A(2, 3), B(6, 3), C(6, 7), and D(2, 7). For the final banner, he 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 of C'. C' = (15, 17.5) Final Answer: (15, 17.5)

  7. A right triangle is drawn on a coordinate plane with vertices at A(0,0), B(6,0), and C(0,8). The triangle is dilated from the point (2,1) with a scale factor of 3. What are the coordinates of the dilated triangle's vertices? Express your answer as three ordered pairs: A', B', C'. Answer: ( -4, -2), (10, -2), (-4, 22) Solution: To dilate from point (2,1), we first translate all points so that (2,1) becomes the origin.
    Full step-by-step solution

    Step 1: To dilate from point (2,1), we first translate all points so that (2,1) becomes the origin. A(0,0) becomes (0-2, 0-1) = (-2,-1) B(6,0) becomes (6-2, 0-1) = (4,-1) C(0,8) becomes (0-2, 8-1) = (-2,7) Step 2: Apply the scale factor of 3 to these translated coordinates: A: (-2,-1) × 3 = (-6,-3) B: (4,-1) × 3 = (12,-3) C: (-2,7) × 3 = (-6,21) Step 3: Translate back by adding (2,1) to each coordinate: A': (-6+2, -3+1) = (-4,-2) B': (12+2, -3+1) = (14,-2) C': (-6+2, 21+1) = (-4,22) Step 4: Verify the answer: A'(-4,-2), B'(14,-2), C'(-4,22) The dilated triangle's vertices are at (-4,-2), (14,-2), and (-4,22).