Dilation definition formula scale factor and solved examples
The concept of dilation in maths is a key transformation in geometry. It helps students understand how shapes resize while maintaining their proportions. This knowledge is essential for school and competitive exams, including JEE and Olympiads, and also appears in real-life applications like maps and art. Mastering dilation makes solving questions on similarity and transformation much easier.
Understanding Dilation in Maths
A dilation is a transformation that alters the size of a figure but not its shape. In geometry, it involves enlarging or reducing a shape with respect to a fixed point, known as the center of dilation. During dilation, each point of the shape moves along a straight line that passes through the center, and the movement is determined by a scale factor. If the scale factor is greater than 1, the figure expands; if it’s between 0 and 1, it contracts.
Key properties of dilation:
- Does not change shape, only size
- All angles remain the same (congruent)
- Corresponding sides remain parallel
- The center and scale factor determine how the figure stretches or shrinks
Dilation Formula and Explanation
Dilation Formula (center (a, b), scale factor k):
Given any point (x, y) in the plane:
Dilation: (x, y) → (k(x – a) + a, k(y – b) + b)
If the center is the origin (0, 0):
Dilation: (x, y) → (k·x, k·y)
Where:
k = Scale factor (k > 1: enlargement, 0 < k < 1: reduction)
(a, b) = Center of dilation
For example, using a scale factor k = 2 centered at (0, 0), the point (3, 5) becomes (6, 10). If k = 0.5, (3, 5) becomes (1.5, 2.5).
Types of Dilation
| Type | Scale Factor Range | Description |
|---|---|---|
| Enlargement | k > 1 | The shape grows bigger. |
| Reduction | 0 < k < 1 | The shape shrinks. |
| No Change | k = 1 | The shape stays the same size (image and original are congruent). |
| Negative Scale Factor | k < 0 | The image is enlarged or reduced and also reflected through the center. |
Worked Examples of Dilation
Let's go through some step-by-step examples.
-
Origin as Center, Enlargement:
Given triangle vertices A(1, 2), B(3, 2), and C(2, 5), perform a dilation with k = 2 centered at (0, 0).- A' = (2×1, 2×2) = (2, 4)
- B' = (2×3, 2×2) = (6, 4)
- C' = (2×2, 2×5) = (4, 10)
-
Center Not at Origin, Reduction:
Dilation of point P(5, 7) with k = 0.5 and center at (2, 3):-
New x: 0.5 × (5 – 2) + 2 = 0.5 × 3 + 2 = 1.5 + 2 = 3.5
New y: 0.5 × (7 – 3) + 3 = 0.5 × 4 + 3 = 2 + 3 = 5
So, P' = (3.5, 5)
-
New x: 0.5 × (5 – 2) + 2 = 0.5 × 3 + 2 = 1.5 + 2 = 3.5
-
Negative Scale Factor:
Dilation of point Q(4, 6) with k = -1, center (0, 0):- Q' = (-1 × 4, -1 × 6) = (-4, -6)
Here, the image is both the same size and reflected through the origin.
Practice Problems
- Dilate the point (2, 3) by a factor of 3 with the center at the origin. What is the image?
- Dilate the point (4, 8) by a factor of 0.25 with the center at the origin.
- If triangle DEF has vertices D(0, 1), E(2, 3), F(4, 1), find the coordinates after dilation by k = 2 with center at (1, 2).
- What happens to the point (5, -2) if dilated with scale factor -2, center (0, 0)?
- If A(1, 7) dilates using k = 0.5 and center (1, 1), what are the new coordinates?
Common Mistakes to Avoid
- Confusing dilation with reflection, rotation, or translation.
- Not applying the scale factor to the distance from the center—not the coordinates directly if center is not at (0, 0).
- Forgetting negative scale factors also reflect the image.
- Miscalculating new points when the center is not the origin.
- Mixing up similarity and dilation: All dilations create similar figures, but not all similar figures are dilations.
Real-World Applications
Dilation is used in many real-life situations, such as:
- Maps and Scaling: Enlarging or reducing drawings to scale (like blueprints, maps, or models).
- Photography and Art: Resizing images while preserving proportions.
- Engineering: Creating scale models of buildings or cars.
- Biology: Comparing scaled diagrams of organs or cells for better understanding.
- Computer Graphics: Digital zooming and shrinking (image processing).
In this topic, we learned about dilation in maths, its definitions, types, the dilation formula, and real-world uses. Understanding dilation makes it easier to solve geometry problems, work with transformations, and recognize similar shapes in math and beyond. At Vedantu, we simplify such geometry concepts to help students grasp them easily and apply them confidently in exams and practical tasks.
FAQs on Understanding the Concept of Dilation in Geometry
1. What is dilation in geometry?
Dilation in geometry is a transformation that changes the size of a figure but keeps its shape the same. It produces a new figure called the image by enlarging or reducing the original figure from a fixed point called the center of dilation.
- All side lengths are multiplied by a scale factor (k).
- Angles remain equal (angle measure is preserved).
- The image and original figure are similar figures.
2. What is the formula for dilation on the coordinate plane?
The formula for dilation about the origin is (x, y) → (kx, ky), where k is the scale factor. In coordinate geometry, each coordinate is multiplied by the same scale factor.
- If k > 1, the figure is enlarged.
- If 0 < k < 1, the figure is reduced.
- If k < 0, the image is reflected through the origin and resized.
3. How do you perform a dilation step by step?
To perform a dilation, multiply each coordinate of the figure by the scale factor from the center of dilation. Follow these steps:
- Identify the center of dilation.
- Determine the scale factor (k).
- Multiply each coordinate by k (if centered at origin).
- Plot the new coordinates to form the image.
4. What is a scale factor in dilation?
The scale factor in dilation is the number (k) that determines how much a figure is enlarged or reduced. It controls the ratio of corresponding side lengths between the image and the original figure.
- k > 1 → enlargement
- 0 < k < 1 → reduction
- k = 1 → no change
5. What is the difference between enlargement and reduction in dilation?
The difference between enlargement and reduction lies in the value of the scale factor. Enlargement happens when k > 1, while reduction occurs when 0 < k < 1.
- Enlargement: Image is bigger than the original (example: k = 3).
- Reduction: Image is smaller than the original (example: k = 0.5).
6. Does dilation change the area and perimeter of a figure?
Dilation changes perimeter by the scale factor k and changes area by the square of the scale factor, k². This means linear measurements and area scale differently.
- New perimeter = k × original perimeter
- New area = k² × original area
7. What happens if the scale factor is negative in dilation?
If the scale factor is negative, the image is reflected through the center of dilation and resized. A negative k reverses the direction of the figure across the center point.
- k = −2 doubles the size and reflects the figure.
- Coordinates are multiplied by the negative value.
8. How do you dilate a figure not centered at the origin?
To dilate a figure not centered at the origin, use the formula (x', y') = (a + k(x − a), b + k(y − b)), where (a, b) is the center of dilation. Steps include:
- Subtract the center coordinates.
- Multiply by k.
- Add the center coordinates back.
9. Why does dilation produce similar figures?
Dilation produces similar figures because it preserves angle measures and keeps side lengths proportional. All corresponding sides are multiplied by the same scale factor k.
- Angles remain equal.
- Side length ratios are constant.
10. Can you give an example of a dilation with a triangle?
A dilation of a triangle multiplies all vertex coordinates by the scale factor to create a similar triangle. Example:
- Original triangle vertices: A(1,1), B(2,1), C(1,3)
- Scale factor k = 2
- New vertices: A′(2,2), B′(4,2), C′(2,6)
