Scale Factor and Dilations

Grade 10 · mathematics · 80 practice problems · read aloud

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Scale Factor and Dilations

📐 What is a Dilation?

A dilation is a transformation that changes the size of a figure but not its shape. The scale factor tells you how much larger or smaller the image becomes.

Why it's useful: Dilations are everywhere! They are used in creating scale models (like maps and blueprints), in photography (zooming in and out), and in computer graphics.

🧮 How to Perform a Dilation

  1. Identify the scale factor (k) and the center of dilation.
  2. Measure the distance from the center of dilation to a vertex of the original shape.
  3. Multiply that distance by the scale factor.
  4. Plot the new vertex along the same line from the center.
  5. Repeat for all vertices and connect them to form the dilated image.

Key Rule: If |k| > 1, the image is an enlargement. If 0 < |k| < 1, the image is a reduction.

🔍 Worked Examples

Example 1: Enlargement

A triangle has vertices at A(2, 1), B(4, 1), C(2, 4). Dilate it from the origin (0,0) with a scale factor of 2.

Step 1: Multiply each coordinate by k=2.
A' = (2*2, 1*2) = (4, 2)
B' = (4*2, 1*2) = (8, 2)
C' = (2*2, 4*2) = (4, 8)
The new triangle is A'(4,2), B'(8,2), C'(4,8). It's twice as big!

Example 2: Reduction

A rectangle has vertices at P(6, 3), Q(12, 3), R(12, 9), S(6, 9). Dilate it from the origin with a scale factor of 1/3.

Step 1: Multiply each coordinate by k=1/3.
P' = (6 * 1/3, 3 * 1/3) = (2, 1)
Q' = (12 * 1/3, 3 * 1/3) = (4, 1)
R' = (12 * 1/3, 9 * 1/3) = (4, 3)
S' = (6 * 1/3, 9 * 1/3) = (2, 3)
The new rectangle is one-third the size of the original.

⚠️ Common Mistakes to Avoid

  • Adding instead of multiplying: A scale factor of 2 means multiply distances by 2, not add 2 to the coordinates.
  • Ignoring the center: If the center of dilation is not the origin, you must measure distances from that specific point.
  • Confusing enlargement/reduction: A scale factor of 1/2 makes the figure smaller, not bigger!

💡 Tips & Tricks

  • Memory Aid: "K is for how many times bigger." K > 1 = bigger, 0 < K < 1 = smaller.
  • Coordinate Shortcut: When the center is the origin (0,0), simply multiply all coordinates by the scale factor.
  • Check Your Work: Corresponding angles in the original and dilated image must be equal. The shape doesn't change!

🎯 How to Practice

  • Graph Paper is Your Best Friend: Draw a simple shape and practice dilating it from different centers with different scale factors.
  • Use Online Tools: Find interactive geometry software to visually check your work.
  • Word Problems: Practice with real-world contexts like finding the actual distance from a map scale.
  • Mix It Up: Try problems that give you the pre-image and image, and ask you to find the scale factor.

Practice problems

6 of the 80, worked through step by step — try them before opening the answer.

1 log₂(8) + log₃(81) = ?

Hint: Remember that logarithms are exponents. For example, log₅(25) = 2 because 5² = 25.

Show the answer

Answer: 7

  1. Evaluate log₂(8). Since 2³ = 8, log₂(8) = 3.
  2. Evaluate log₃(81). Since 3⁴ = 81, log₃(81) = 4.
  3. Add the results: 3 + 4 = 7.

The answer is 7.

2 √(x² - 6x + 9) = ? when x = 4

Hint: This expression can be rewritten as a perfect square. Consider what happens when you take the square root of a squared expression.

Show the answer

Answer: 1

  1. Substitute x = 4 into the expression: √(4² - 6×4 + 9)
  2. Calculate inside the square root: 16 - 24 + 9 = 1
  3. Take the square root: √1 = 1

The answer is 1.

3 Dilate point (4,6) by scale factor 2 from origin

Hint: To dilate a point from the origin, multiply each coordinate by the scale factor. For example, dilating (2,4) by scale factor 3 would give you (6,12).

Show the answer

Answer: (8,12)

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

The answer is (8,12).

4 Dilate point (5,10) by scale factor 2 from origin

Hint: Multiply each coordinate by the scale factor when the center of dilation is the origin

Show the answer

Answer: (10,20)

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

The answer is (10,20).

5 Dilate point (7,12) by scale factor 2 from origin

Hint: To dilate a point from the origin, multiply each coordinate by the scale factor. For example, dilating (3,5) by scale factor 4 would give you (12,20).

Show the answer

Answer: (14,24)

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

The answer is (14,24).

6 Dilate point (8, 10) by scale factor 2 from origin

Hint: To dilate a point from the origin, multiply each coordinate by the scale factor. For example, dilating (3, 5) by scale factor 4 would give you (12, 20).

Show the answer

Answer: (16, 20)

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

The answer is (16, 20).

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