

How Do You Perform Dilation of a Shape Using a Scale Factor?
The concept of dilation in maths plays a key role in geometry and transformations. It describes how a shape can be enlarged or reduced from a fixed point, without changing its overall proportions or the measures of its angles. You’ll regularly use dilation in coordinate geometry, similar triangles, and real-life mapping or modeling situations.
What Is Dilation in Maths?
A dilation in maths is a type of transformation that resizes a figure by expanding or shrinking it, based on a scale factor, from a fixed point called the center of dilation. The shape remains similar to its original — angles stay the same, and each side changes in length but not in proportion. You’ll find this concept applied in scaling, map readings, and model-making.
Key Formula for Dilation in Maths
Here’s the standard formula:
For a point \( P(x, y) \) and center \( C(a, b) \), with scale factor \( k \):
New Point: \( (x', y') = (a + k \cdot (x-a), \; b + k \cdot (y-b)) \)
If the center is the origin \((0,0)\), this simplifies to:
\( (x', y') = (k \cdot x, k \cdot y) \)
Types of Dilation
Type | Scale Factor (k) | Effect |
---|---|---|
Enlargement | k > 1 | Image is larger than original figure |
Reduction | 0 < k < 1 | Image is smaller than original figure |
Same Size (Identity) | k = 1 | Image and original figure are congruent |
Reflectional Dilation | k < 0 | Image is inverted (flipped) and resized |
Step-by-Step Illustration
- Identify the center of dilation and scale factor.
Example: Center at (0, 0), scale factor \( k = 1/2 \). - For each point, subtract the center, multiply by scale factor, then add the center.
Original point A(6, 4):
New x = \( 0 + \frac{1}{2} \cdot (6-0) = 3 \)
New y = \( 0 + \frac{1}{2} \cdot (4-0) = 2 \)
So, image A' is (3, 2). - Repeat for other points of the shape.
If B(2, 6), then B' = (1, 3). - Connect the image points to get the dilated figure.
Dilation Example Problems
Problem 1: Dilate point (4, 6) with center at origin and scale factor 2.
1. Center at (0, 0), k = 2
2. New x = 0 + 2 × (4−0) = 8
3. New y = 0 + 2 × (6−0) = 12
Answer: (8, 12)
Problem 2: A triangle has vertices A(3, 2), B(5, 4), C(1, 2). Find the image when dilated from origin with scale factor 0.5.
1. For A: (3,2) → (0.5×3, 0.5×2) = (1.5, 1)
2. For B: (5,4) → (2.5, 2)
3. For C: (1,2) → (0.5, 1)
Image vertices: A'(1.5,1), B'(2.5,2), C'(0.5,1)
Frequent Errors and Misunderstandings
- Mixing up enlargement (k>1) with reduction (k<1).
- Applying scale factor to angles (angles do not change in dilation).
- Forgetting to use the center of dilation when it’s not at the origin.
- Confusing image (after dilation) with preimage (original shape).
Relation to Other Concepts
The idea of dilation in maths connects closely with similar figures and scale factor. Dilation is also a key part of geometry transformations and is commonly asked in coordinate geometry exam problems.
Real-World & Exam Applications
Dilation is not just useful in Maths but also in drawing scale maps, making models, and creating blueprints. In exam questions for class 8 to 12, dilation often appears in geometry, coordinate transformations, and even in questions about scaling patterns or figures. Being able to perform dilation step by step is a must-have skill for competitive exams like Olympiads and JEE. Vedantu offers live classes and interactive resources that make learning dilation easy and practical.
Try These Yourself
- Dilate the point (5, 10) with center at origin and scale factor 0.4. What is the image?
- A square with vertices at (2,2), (2,4), (4,4), (4,2) is dilated from origin with k = 3. Find the new coordinates.
- Dilate triangle with vertices (0,0), (2,2), (4,0) using center (0,0) and k = −1.
Classroom Tip
A quick way to remember dilation: “Angles remain, sides scale!” Meaning, no matter how you dilate a shape, the angles don’t change, but sides do—by the scale factor. Use graph paper or digital geometry tools to practice dilations step by step, just like Vedantu’s teachers use in live classes.
We explored dilation in maths: its definition, formula, solved examples, real-life connections, and clever tips. Continue practicing these skills with Vedantu and explore more about transformations and coordinate geometry to become confident in any Maths exam!
FAQs on Dilation in Maths: Meaning, Formula & Applications
1. What is meant by dilation in mathematics?
In mathematics, dilation is a type of transformation that changes the size of a figure without altering its shape or orientation. The result of a dilation is a new figure that is geometrically similar to the original. This process either enlarges (makes bigger) or reduces (makes smaller) the figure based on a specific scale factor.
2. What is the role of the scale factor and the center of dilation?
The two key components in a dilation are:
- Scale Factor (k): This is a number that determines the amount of resizing. If the scale factor is greater than 1, the figure is enlarged. If it is between 0 and 1, the figure is reduced.
- Center of Dilation: This is a fixed point on the plane from which the dilation is performed. All points on the original figure are scaled in relation to their distance from this center point.
3. What is the difference between an enlargement and a reduction in dilation?
An enlargement occurs when the scale factor (k) is greater than 1 (k > 1). The resulting image is larger than the original pre-image. A reduction occurs when the scale factor is a positive number less than 1 (0 < k < 1), resulting in an image that is smaller than the pre-image.
4. What is the formula to find the coordinates of a dilated shape?
To find the new coordinates (x', y') of a point (x, y) after dilation, you use the formula: (x', y') = (k(x - a) + a, k(y - b) + b). In this formula, (x, y) is the original point, (a, b) is the center of dilation, and 'k' is the scale factor. If the center of dilation is the origin (0, 0), the formula simplifies to (x', y') = (kx, ky).
5. What mathematical operation is primarily used to perform a dilation?
The primary mathematical operation used for dilation is multiplication. The distance of each point from the center of dilation is multiplied by the scale factor to determine the location of the new, corresponding point in the dilated image.
6. What are some real-world examples of dilation?
Dilation is a common concept found in many real-world applications. Examples include:
- Photography: Zooming in or out with a camera lens dilates the image.
- Architecture: Blueprints are scaled-down reductions of actual buildings.
- Cartography: Maps are reductions of large geographical areas.
- Graphic Design: Resizing an image on a computer is a direct application of dilation.
7. How is dilation different from other geometric transformations like translation or rotation?
Dilation is fundamentally different from transformations like translation, rotation, and reflection. While translation (sliding), rotation (turning), and reflection (flipping) are rigid transformations that preserve the size and shape of a figure, dilation is a non-rigid transformation. It preserves the shape but changes the size, unless the scale factor is 1.
8. How does dilating a 2D shape affect its perimeter and area?
When a 2D shape is dilated by a scale factor 'k', its properties change predictably:
- The new perimeter is the original perimeter multiplied by the scale factor (k).
- The new area is the original area multiplied by the square of the scale factor (k²).
For example, doubling the side lengths of a square (k=2) doubles its perimeter but quadruples its area.
9. What is the relationship between dilation and geometric similarity?
Dilation is the transformation that formally defines similarity. Two figures are considered similar if one can be mapped onto the other through a sequence of rigid transformations (like rotation or translation) and a dilation. In simpler terms, dilation is the specific action of scaling that creates similar figures.
10. What happens if the scale factor for a dilation is negative or exactly 1?
These are special cases of dilation:
- If the scale factor (k) is 1, the dilation results in an image that is identical in size and position to the original. This is known as an identity transformation.
- If the scale factor (k) is negative, the dilation performs two actions: it scales the figure by the absolute value of k and also rotates it 180 degrees around the center of dilation.
11. Why is dilation considered a non-rigid transformation?
Dilation is a non-rigid transformation because it does not preserve the distance between points, meaning it changes the size of the figure. Rigid transformations, also known as isometries, must maintain all distances and angle measures. Since dilation alters side lengths (unless the scale factor is 1), it changes the figure's size and therefore cannot be rigid.











