Scale Factor and Dilations

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Answer Key & Explanations

Scale Factor and Dilations · Grade 10 · Worksheet 3

1. Dilate point (4, 8) by scale factor 2 from center (0, 0) Answer: (8, 16) Solution: Identify the original point coordinates: (4, 8) Identify the scale factor: 2 Multiply the x-coordinate by the scale factor: 4 × 2 = 8 Multiply the y-coordinate by the scale factor: 8 × 2 = 16 Write the new coordinates: (8, 16) The answer is (8, 16).
   Full step-by-step solution

   Step 1: Identify the original point coordinates: (4, 8) Step 2: Identify the scale factor: 2 Step 3: Multiply the x-coordinate by the scale factor: 4 × 2 = 8 Step 4: Multiply the y-coordinate by the scale factor: 8 × 2 = 16 Step 5: Write the new coordinates: (8, 16) The answer is (8, 16).

2. An architect is designing a new building and creates a scale model where the model's height is 1.5 meters. The actual building will be 45 meters tall. If the volume of the model is 0.8 cubic meters, what will be the volume of the actual building in cubic meters? Answer: 21600 Solution: The model's height is 1.5 m, and the actual building's height is 45 m.
   Full step-by-step solution

   Let's go step-by-step. --- **Step 1: Understand the scale factor for length** The model's height is 1.5 m, and the actual building's height is 45 m. The **length scale factor** \( k \) is: \[ k = \frac{\text{Actual height}}{\text{Model height}} = \frac{45}{1.5} \] \[ k = 30 \] So every length in the actual building is 30 times the corresponding length in the model. --- **Step 2: Relate length scale factor to volume scale factor** Volume scales with the **cube** of the length scale factor. \[ \text{Volume scale factor} = k^3 = 30^3 \] \[ 30^3 = 30 \times 30 \times 30 = 900 \times 30 = 27000 \] --- **Step 3: Find the actual building's volume** Model volume = 0.8 cubic meters. Actual volume = Model volume × Volume scale factor \[ \text{Actual volume} = 0.8 \times 27000 \] \[ 0.8 \times 27000 = 0.8 \times (27000) = 21600 \] --- **Step 4: Final answer** \[ \boxed{21600} \] The actual building's volume will be **21600 cubic meters**.

3. Dilate point (4,6) by scale factor 2 from origin Answer: (8,12) Solution: Identify the original point coordinates: (4,6) Identify the scale factor: 2 Multiply the x-coordinate by the scale factor: 4 × 2 = 8 Multiply the y-coordinate by the scale factor: 6 × 2 = 12 Write the new coordinates: (8,12) The answer is (8,12).
   Full step-by-step solution

   Step 1: Identify the original point coordinates: (4,6) Step 2: Identify the scale factor: 2 Step 3: Multiply the x-coordinate by the scale factor: 4 × 2 = 8 Step 4: Multiply the y-coordinate by the scale factor: 6 × 2 = 12 Step 5: Write the new coordinates: (8,12) The answer is (8,12).

4. Matiu is a digital artist creating a geometric design. He plots a triangle with vertices at A(-14, 22), B(26, -18), and C(10, 34) on his coordinate canvas. He wants to enlarge the triangle by applying a dilation centered at the origin with a scale factor of 0.75. What are the coordinates of vertex A' after this dilation? Answer: (-10.5, 16.5) Solution: Identify the original coordinates of vertex A: (-14, 22) The dilation is centered at the origin with a scale factor of 0.75.
   Full step-by-step solution

   Step 1: Identify the original coordinates of vertex A: (-14, 22) Step 2: The dilation is centered at the origin with a scale factor of 0.75. Apply the dilation formula: (x', y') = (k * x, k * y) where k = 0.75. Step 3: Multiply the x-coordinate: x' = 0.75 * (-14) = -10.5 Step 4: Multiply the y-coordinate: y' = 0.75 * 22 = 16.5 Step 5: The coordinates of vertex A' after dilation are (-10.5, 16.5). The answer is (-10.5, 16.5).

5. A kite is drawn on a coordinate plane with vertices at A(-8, 0), B(0, 6), C(8, 0), and D(0, -2). Mere dilates this kite from the origin using a scale factor of 1/4. What are the coordinates of the dilated kite's vertices? Express your answer as four ordered pairs: A', B', C', D'. Answer: (-2, 0), (0, 3/2), (2, 0), (0, -1/2) Solution: The original vertices are A(-8, 0), B(0, 6), C(8, 0), and D(0, -2). Dilation from the origin with scale factor k = 1/4 means each coordinate is multiplied by 1/4.
   Full step-by-step solution

   Step 1: The original vertices are A(-8, 0), B(0, 6), C(8, 0), and D(0, -2). Step 2: Dilation from the origin with scale factor k = 1/4 means each coordinate is multiplied by 1/4. Step 3: Calculate A': (-8 * 1/4, 0 * 1/4) = (-2, 0) Step 4: Calculate B': (0 * 1/4, 6 * 1/4) = (0, 6/4) = (0, 3/2) Step 5: Calculate C': (8 * 1/4, 0 * 1/4) = (2, 0) Step 6: Calculate D': (0 * 1/4, -2 * 1/4) = (0, -2/4) = (0, -1/2) Step 7: The dilated kite's vertices are A'(-2, 0), B'(0, 3/2), C'(2, 0), D'(0, -1/2).

6. A marine biologist is studying coral polyps under a microscope. The original coral polyp image appears as a triangle with vertices at A(-4, 2), B(2, 6), and C(0, -2). When she switches to a higher magnification, the image undergoes a dilation centered at the origin with a scale factor of 3. What are the coordinates of vertex B' after this dilation? Answer: (6, 18) Solution: Identify the original coordinates of vertex B: (2, 6) Apply the dilation formula for center at origin: (x', y') = (k*x, k*y) where k is the scale factor Multiply each coordinate by the scale factor of 3: x' = 3 * 2 = 6, y' = 3 * 6 = 18 The coordinates of B' after dilation are (6, 18) The answer…
   Full step-by-step solution

   Step 1: Identify the original coordinates of vertex B: (2, 6) Step 2: Apply the dilation formula for center at origin: (x', y') = (k*x, k*y) where k is the scale factor Step 3: Multiply each coordinate by the scale factor of 3: x' = 3 * 2 = 6, y' = 3 * 6 = 18 Step 4: The coordinates of B' after dilation are (6, 18) The answer is (6, 18).

7. A polygon is drawn on a coordinate plane with vertices at A(-16, 12), B(20, 12), C(28, -4), D(8, -20), and E(-12, -4). Tane dilates this polygon from the point (4, -8) using a scale factor of 1.5. What are the coordinates of the dilated polygon's vertices? Express your answer as five ordered pairs: A', B', C', D', E'. Answer: (-26, 22), (28, 22), (40, -2), (10, -26), (-20, -2) Solution: Center of dilation is (4, -8). Translate each vertex so that (4, -8) becomes the origin by subtracting (4, -8) from each coordinate.
   Full step-by-step solution

   Step 1: Center of dilation is (4, -8). Translate each vertex so that (4, -8) becomes the origin by subtracting (4, -8) from each coordinate. A(-16, 12) becomes (-16-4, 12-(-8)) = (-20, 20) B(20, 12) becomes (20-4, 12-(-8)) = (16, 20) C(28, -4) becomes (28-4, -4-(-8)) = (24, 4) D(8, -20) becomes (8-4, -20-(-8)) = (4, -12) E(-12, -4) becomes (-12-4, -4-(-8)) = (-16, 4) Step 2: Apply scale factor 1.5 to each translated coordinate: A: (-20 * 1.5, 20 * 1.5) = (-30, 30) B: (16 * 1.5, 20 * 1.5) = (24, 30) C: (24 * 1.5, 4 * 1.5) = (36, 6) D: (4 * 1.5, -12 * 1.5) = (6, -18) E: (-16 * 1.5, 4 * 1.5) = (-24, 6) Step 3: Translate back by adding (4, -8) to each result: A': (-30 + 4, 30 + (-8)) = (-26, 22) B': (24 + 4, 30 + (-8)) = (28, 22) C': (36 + 4, 6 + (-8)) = (40, -2) D': (6 + 4, -18 + (-8)) = (10, -26) E': (-24 + 4, 6 + (-8)) = (-20, -2) The dilated polygon's vertices are A'(-26, 22), B'(28, 22), C'(40, -2), D'(10, -26), E'(-20, -2).

8. Olivia dilates point (7, 9) by scale factor 3 from center (1, 1). Find the new coordinates. Answer: (19, 25) Solution: Translate the point so the center of dilation becomes the origin. Subtract the center coordinates from the point coordinates: (7 - 1, 9 - 1) = (6, 8) Apply the scale factor to the translated coordinates: (6 × 3, 8 × 3) = (18, 24) Translate back by adding the center coordinates: (18 + 1, 24 + 1)…
   Full step-by-step solution

   Step 1: Translate the point so the center of dilation becomes the origin. Subtract the center coordinates from the point coordinates: (7 - 1, 9 - 1) = (6, 8) Step 2: Apply the scale factor to the translated coordinates: (6 × 3, 8 × 3) = (18, 24) Step 3: Translate back by adding the center coordinates: (18 + 1, 24 + 1) = (19, 25) The answer is (19, 25).