How Does Dilation Work in Geometry?
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 Dilation in Maths: Definition, Properties & Examples
1. What is the concept of dilation in mathematics?
Dilation is a geometric transformation that resizes a figure, enlarging or reducing it proportionally. It's defined by a center of dilation and a scale factor. The scale factor determines the size change; a scale factor greater than 1 enlarges the figure, while a scale factor between 0 and 1 reduces it. The center of dilation is the fixed point around which the transformation occurs.
2. What are the three types of dilation?
While not strictly categorized into three types, dilations are typically described based on the scale factor:
- Enlargement: Scale factor greater than 1 (figure gets bigger).
- Reduction: Scale factor between 0 and 1 (figure gets smaller).
- Dilation with center not at the origin: This involves a more complex formula, accounting for the shift in the center of dilation.
3. What is the dilation formula if the center is not at the origin?
The dilation formula for a point (x, y) with center (a, b) and scale factor k is: D(x, y) → (k(x-a)+a, k(y-b)+b). This formula accounts for the shift from the origin. If the center is at the origin (0,0), the formula simplifies to D(x, y) → (kx, ky).
4. How do you find the scale factor in dilation?
The scale factor (k) in dilation is the ratio of the corresponding lengths in the image and pre-image. Measure a corresponding length in both the original figure (pre-image) and the dilated figure (image), and then divide the length of the image by the length of the pre-image to find k. For example, if a line segment of length 5 is dilated to a length of 15, the scale factor is 15/5 = 3.
5. What is an example of dilation?
Enlarging a photograph is a real-world example of dilation. The original photo is the pre-image, and the enlarged version is the image. The center of dilation might be the center of the photo, and the scale factor depends on the degree of enlargement.
6. Can dilation change the orientation of a shape?
A positive scale factor preserves the orientation of a shape. However, a negative scale factor will result in a reflection across the center of dilation in addition to resizing, thus changing the orientation.
7. What is the dilation rule formula?
The dilation rule depends on the location of the center of dilation. If the center is at the origin (0,0), the rule is (x, y) → (kx, ky) where k is the scale factor. If the center is at a point (a, b), the rule becomes (x, y) → (k(x-a) + a, k(y-b) + b). This ensures correct scaling around a chosen point.
8. What refers to dilation?
Dilation refers to the geometric transformation that changes the size of a figure without changing its shape. The transformation is defined by a center point and a scale factor that determines the amount of enlargement or reduction.
9. What is the center of dilation?
The center of dilation is the fixed point around which a figure is enlarged or reduced during a dilation. All points on the figure are scaled proportionally with respect to this fixed center point.
10. How is dilation used in real life?
Dilation has many real-world applications, including:
- Mapmaking: Creating scaled-down representations of geographical areas.
- Engineering: Designing and scaling blueprints for buildings or machines.
- Photography: Zooming in or out on images.
- Image processing: Resizing digital images.
