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 dilation in Maths?
Dilation in mathematics is a transformation that changes the size of a geometric figure but preserves its shape. It's performed by multiplying the coordinates of each point by a scale factor, relative to a center of dilation. A scale factor greater than 1 results in enlargement, while a scale factor between 0 and 1 results in reduction.
2. How do you perform dilation using a scale factor?
To perform a dilation with a scale factor k and center of dilation (a, b):
- Identify the center of dilation: This is the fixed point around which the shape will be enlarged or reduced.
- Multiply the coordinates: For each point (x, y) in the original figure, calculate the new coordinates (x', y') using the formula: x' = k(x - a) + a and y' = k(y - b) + b.
- Plot the new points: Connect the new points to form the dilated figure.
3. What is the center of dilation?
The center of dilation is a fixed point in the plane. All points in the original figure are scaled relative to this point. The distance from the center of dilation to each point in the original figure is multiplied by the scale factor to determine the corresponding point's location in the dilated figure.
4. What happens if the scale factor is less than 1?
If the scale factor is less than 1 (but greater than 0), the dilation results in a reduction. The dilated figure will be smaller than the original figure, but it will retain the same shape and proportions.
5. What is the formula for dilation in coordinate geometry?
The formula for dilation in coordinate geometry, with center (a, b) and scale factor k, is: (x', y') = (k(x - a) + a, k(y - b) + b), where (x, y) are the coordinates of a point in the original figure, and (x', y') are the coordinates of the corresponding point in the dilated figure.
6. How does dilation affect the area or perimeter of a shape?
Dilation affects the area and perimeter proportionally to the square of the scale factor. If the scale factor is k, the new area is k² times the original area, and the new perimeter is k times the original perimeter.
7. Can two different shapes be dilation images of each other?
No, two different shapes cannot be dilation images of each other unless one is a scaled version of the other. Dilation only changes the size, not the shape, so the figures must be similar.
8. Is dilation a congruence transformation? Why or why not?
No, dilation is not a congruence transformation. Congruence transformations preserve both size and shape. Dilation only preserves shape; it changes the size unless the scale factor is 1.
9. What are real-life examples where dilation is used outside Maths?
Dilation is used in various real-world applications, including: creating maps (scaling down geographical areas), designing blueprints for buildings (scaling up architectural plans), and enlarging or reducing images in graphic design software.
10. How do dilation errors typically occur in student answers?
Common errors include: incorrect application of the scale factor, misidentification of the center of dilation, and inaccurate calculations of new coordinates. Students may also struggle to visualize the transformation and accurately plot the new points.
11. What is the difference between dilation and similarity?
While dilation is a *type* of transformation that creates similar figures, similarity is a broader concept. Two figures are similar if one can be obtained from the other through a sequence of transformations including dilation, rotation, reflection, and translation. Dilation only involves scaling.
