How to Find the Scale Factor of a Dilation
Introduction
Dilation is a geometric transformation that alters the size of an object while preserving its shape. This transformation involves resizing any two-dimensional or three-dimensional shape while retaining its original shape. It is a transformation that makes the objects bigger or smaller with respect to the original object.
The scale factor determines the magnitude of the change in size, with values greater than `1` indicating enlargement and values between `0` and `1` indicating reduction. Dilation is commonly used in geometry to resize figures without altering their proportions, and it plays a significant role in various fields such as mapmaking, computer graphics, and architectural design.
Definition of Dilation
In geometry, dilation is a transformation that enlarges or reduces the size of a figure without altering its shape. It involves stretching or shrinking the figure in all directions from a fixed point called the center of dilation. Each point on the object being dilated moves towards or away from the center of dilation while maintaining the same proportionate relationship to the center. The center of dilation may be located inside, outside or on the figure that is dilated. The scale factor determines the amount of enlargement or reduction.
It’s good to know the following terminologies that go with definition of dilation in math.
`1`. Components of Dilation:
- Pre-image: The original object before dilation.
- Image: The resulting object after dilation.
`2`. Types of Dilation:
- Expansion: Enlarges the size of the object.
- Contraction: Reduces the size of the object.
`3`. Scale Factor:
The scale factor is a numerical value that determines the extent of enlargement or reduction during dilation.
- If the scale factor is greater than `1`, it indicates expansion.
- If the scale factor is between `0` and `1`, it indicates contraction.
Understanding dilation math definition is fundamental in geometry as it allows for the transformation of objects while preserving their fundamental characteristics, particularly their shape.
Example : Consider a square with side length `5` units that has been dilated with a scale factor of `3`. What is the side length of the resulting image?
Solution:
If it is dilated with a scale factor of `3`, the resulting image will have a side length of ` 3 \times 5 ` which is ` 15 ` `\text{units}`. This represents expansion because the scale factor is greater than `1`.
The Scale Factor
`1`. Definition:
The scale factor is a ratio that determines the proportional change in size between the pre-image (original figure) and the image (resulting figure) in a dilation.
`2`. Role in Dilation:
The scale factor indicates how much the dimensions of the pre-image are multiplied or divided to obtain the dimensions of the image.
It governs the magnitude of enlargement or reduction in the figure during dilation.
`3`. Calculation:
To calculate the scale factor, divide the length of any side (or dimension) of the image by the corresponding length of the pre-image.
`\text{Scale Factor} = \frac{\text{Length of Image}}{\text{Length of Pre-image}}`
`4`. Interpretation:
A scale factor greater than `1` indicates enlargement, where the image is larger than the pre-image.
A scale factor between `0` and `1` indicates reduction, where the image is smaller than the pre-image.
A scale factor of `1` indicates no change in size, where the image is identical to the pre-image.
`5`. Relationship with Dilations:
In dilation, every point in the pre-image is stretched or shrunk away from or towards the center of dilation by the same scale factor.
The scale factor is uniform across the entire figure, ensuring that all lengths, angles, and shapes are preserved proportionally.
Scale factors are used in various fields such as architecture, engineering, mapmaking, and computer graphics to resize objects while preserving their proportions. They are essential for accurately representing real-world objects at different scales.
Scale Factor Formula
The dilation formula to calculate the scale factor is straightforward:
`\text{Scale Factor} = \frac{\text{Length of Image}}{\text{Length of Pre-image}}`
This formula for dilation applies to any linear dimension of the figure being dilated, such as side lengths, heights, or radii.
This scale factor formula can be rearranged to solve for different variables.
Rearranged Formula:
`\text{Dimension of the new shape} = \text{Dimension of the original shape} \times \text{Scale Factor}`
This rearranged formula is handy when you know the original shape and its dimension(s), as well as the scale factor, and you want to find out the dimensions of the dilated shape.
Example `1`: If a square with a side length of `5` `\text{units}` is dilated to a square with a side length of `12.5` `\text{units}`, what is the scale factor of dilation?
Solution:
`\text{Scale Factor} = \frac{\text{Length of Image}}{\text{Length of Pre-image}}`
`\text{Scale Factor} = \frac{12.5}{5}`
`\text{Scale Factor} = 2.5`
Hence the scale factor of dilation is `2.5` indicating an enlargement by a factor of `2.5`.
Example `2`: A circle with a radius of `10` `\text{units}` is dilated to a circle with a radius of `5` `\text{units}`. What is the scale factor of dilation?
Solution:
`\text{Scale Factor} = \frac{\text{Length of Image}}{\text{Length of Pre-image}}`
`\text{Scale Factor} = \frac{5}{10}`
\( \text{Scale Factor} = 0.5 \)
Hence the scale factor of dilation is `0.5` indicating a reduction by a factor of `0.5`.
Finding Scale Factor of Dilation Using Coordinates
Here is explanation of how to calculate the scale factor in dilation using the coordinates of the original and dilated vertices. Let's summarize the steps:
Coordinate Comparison Method
- Take any vertex of the original figure and its corresponding vertex in the dilated figure.
- Compare the coordinates of the original vertex `(x, y)` to the coordinates of the dilated vertex `(x', y')`.
- Calculate the scale factor by dividing each coordinate of the dilated vertex by the corresponding coordinate of the original vertex.
The resulting scale factor will be the same for both `x` and `y` coordinates.
Example: Calculate the scale factor when quadrilateral `MNOP` is reduced to quadrilateral `M'N'O'P'`.
Solution:
The vertices of the original quadrilateral `MNOP` are given by: `M(3,3), N(9,-3), O(-6,-9), P(-6,6)`
After dilation, the corresponding vertices of the dilated quadrilateral `M'N'O'P'` are: `M'(1,1), N'(3,-1), O'(-2,-3), P’(-2,2)`
To find the scale factor, consider the coordinates of any vertex and the coordinates of its corresponding dilated vertex.
Lets consider `N(9,-3)` and `N'(3,-1)`.
On dividing each coordinate of `N'` by the corresponding coordinate of the original vertex `N`, we get
`\text{Scale Factor} = \frac{3}{9} = \frac{1}{3}` and
`\text{Scale Factor} = \frac{-1}{-3} = \frac{1}{3}`
Hence the quadrilateral `MNOP` is scaled down by `1/3` to form the quadrilateral `M'N'O'P'`.
This example illustrates how the coordinates of each vertex are transformed according to the dilation process while maintaining the shape and angles of the original triangle.
Center of Dilation
Let's look into the concept of the center of dilation
`1`. Definition:
The center of dilation is a fixed point in the plane or space around which a geometric figure is enlarged or reduced during dilation.
`2`. Role in Dilation:
The center of dilation serves as the reference point from which the dilation occurs.
All points of the object move away from or towards the center of dilation while maintaining the same proportional relationships.
`3`. Location:
In two-dimensional geometry, the center of dilation is typically specified as a point with coordinates `(x, y)`.
In three-dimensional geometry, it can be a point with coordinates `(P, Q, S)` or even a line. Figure showing the center of dilation at point `R`.
`4`. Determining the Center:
The center of dilation can be explicitly given in a problem or determined based on the context of the dilation.
It's often indicated by a labeled point on a coordinate plane or described relative to other geometric elements.
`5`. Effect on Dilation:
Changing the location of the center of dilation affects how the dilation alters the size of the object.
- If the center of dilation is outside the object, it may appear to stretch or compress in various directions.
- If the center of dilation is inside the object, it may appear to expand outward or contract inward.
For instance:
If a circle is dilated with the center of dilation at its center, the resulting image will be a concentric circle with a changed radius.
If a triangle is dilated with the center of dilation outside the triangle, the image will appear stretched or compressed relative to the center.
`6`. Properties:
All lines drawn from the center of dilation to points on the object are dilated in the same proportion.
The center of dilation is an invariant point during dilation; it remains fixed while other points move.
Understanding the center of dilation is crucial in geometric transformations, particularly in dilation, as it governs how the object is resized and repositioned relative to this central point.
Real-Life Applications
`1`. Map Making: Cartographers use dilation to create maps of different scales. By dilating geographical features such as roads, rivers, and cities, mapmakers can represent vast areas on small-scale maps while preserving relative distances and proportions.
`2`. Architectural Design: Architects use dilation to scale down or enlarge building plans while preserving the architectural details and proportions. This is crucial for creating blueprints, models and prototypes that accurately represent the final structure.
`3`. Art and Design: Artists and designers often use dilation to create images with varying sizes while maintaining their artistic integrity. For example, a graphic designer may use dilation to resize a logo for different applications, such as on a business card or a billboard.
`4`. Photography and Image Editing: In photography and image editing software, dilation is used to resize and manipulate images without distorting the objects within them. This allows photographers and graphic designers to adjust the size and composition of images while preserving their visual integrity.
`5`. Medical Imaging: In medical imaging, dilation is used to enlarge or reduce images of organs, tissues, and structures within the body. This allows doctors and radiologists to examine detailed images of internal structures while maintaining their anatomical proportions.
`6`. Scaling Models: Engineers and model makers use dilation to create scaled-down or scaled-up models of objects and structures. By applying a consistent scale factor, they can accurately represent complex systems while working within limited physical space.
`7`. Computer Graphics and Animation: Dilation is fundamental in computer graphics and animation for resizing and transforming digital objects and characters. This is essential for creating realistic animations and visual effects in movies, video games, and virtual simulations.
`8`. Urban Planning: Urban planners use dilation to create scaled-down models of cities and urban environments for analysis and simulation. This allows them to study the impact of proposed developments and infrastructure projects on the surrounding area.
Solved Examples
Example `1`: Consider a triangle `ABC` with vertices `A(1, 3), B(3, 1),` and `C(1, 1)`. Dilate the triangle with a scale factor of `2` with center of dilation at the origin.
Solution:
To dilate the triangle with a scale factor of `2` and center of dilation at the origin, we need to multiply the coordinates of each vertex by the scale factor.
For vertex `A`:
- Original coordinates: `A(1, 3)`
- Dilation: `A'(2, 6)` `[`Coordinates are doubled: `1 * 2 = 2, 3 * 2 = 6``]`
For vertex `B`:
- Original coordinates: `B(3, 1)`
- Dilation: `B'(6, 2)` `[`Coordinates are doubled: `3 * 2 = 6, 1 * 2 = 2``]`
For vertex `C`:
- Original coordinates: `C(1, 1)`
- Dilation: `C'(2, 2)` `[`Coordinates are doubled: `1 * 2 = 2, 1 * 2 = 2``]`
So, the dilated triangle is `A'B'C'` with vertices `A'(2, 6), B'(6, 2),` and `C'(2, 2)`.
Example `2`: A circle with a radius of `5` `\text{units}` is dilated with a scale factor of `2` and centered at the origin. Find the equation of the dilated circle.
Solution:
Given that the original circle has a radius of `5` `\text{units}` and is centered at the origin `(0, 0)`, its equation is `x^2 + y^2 = 25`, where `x` and `y` are the coordinates of any point on the circle.
To dilate the circle with a scale factor of `2`, we need to double the distance of each point on the original circle from the origin. This results in a new circle with a radius of `2 \times 5 = 10` `\text{units}`.
So, the equation of the dilated circle is `x^2 + y^2 = 100`.
Example `3`: A triangle has vertices `A(2, 4), B(6, 4),` and `C(4, 2)`. If it is dilated with a scale factor of `3` and centered at the origin, find the coordinates of the vertices of the dilated triangle.
Solution:
To dilate the triangle with a scale factor of `3` and centered at the origin, we need to multiply the coordinates of each vertex by the scale factor.
For vertex `A`:
- Original coordinates: `A(2, 4)`
- Dilation: `A'(6, 12)` `[`Coordinates are tripled: `2 * 3 = 6, 4 * 3 = 12``]`
For vertex `B`:
- Original coordinates: `B(6, 4)`
- Dilation: `B'(18, 12)` `[`Coordinates are tripled: `6 * 3 = 18, 4 * 3 = 12``]`
For vertex `C`:
- Original coordinates: `C(4, 2)`
- Dilation: `C'(12, 6)` `[`Coordinates are tripled: `4 * 3 = 12, 2 * 3 = 6``]`
So, the coordinates of the vertices of the dilated triangle are `A'(6, 12), B'(18, 12),` and `C'(12, 6)`.
Example `4`: Figure `A` has been reduced to figure `B`. By what factor has the original figure `A` been scaled down?
Solution:
The length of the Figure `A` (pre-image) is `3` `\text{units}` and the width is `6` `\text{units}`.
The length of the Figure `B` (image) is `1` `\text{unit}` and the width is `2` `\text{units}`.
To find the scale factor, we use the scale formula:
`\text{Scale Factor} = \frac{\text{Length of Image}}{\text{Length of Pre-image}}`
`\text{Scale Factor} = \frac{1}{3}`
Figure `A` has been scaled down by a factor of ` \frac{1}{3} ` to form the Figure `B`.
Example `5`: The length of each side of a rhombus is `12` `\text{units}`. What will be length of each side of the dilated rhombus if the scale factor is `1.5`?
Solution:
Side length of the rhombus `= 12` `\text{units}`
Scale factor `= 1.5`
As per the rearranged formula for scale factor
\( \text{Dimension of the new shape} = \text{Dimension of the original shape} \times \text{Scale Factor} \)
Substituting the values in the formula:
\( \text{Dimension of the new shape} = 12 \times 1.5 = 18\)
Therefore, the side length of the dilated rhombus will be `18` `\text{units}`.
Practice Problems
Q`1`. A rectangle has dimensions `4` `\text{units}` by `6` `\text{units}`. If it is dilated with a scale factor of `0.5` and centered at the origin, find the dimensions of the dilated rectangle.
- `2` `\text{units}` by `3` `\text{units}`
- `8` `\text{units}` by `12` `\text{units}`
- `4.5` `\text{units}` by `6.5` `\text{units}`
- `3` `\text{units}` by `5` `\text{units}`
Answer: a
Q`2`. A square has vertices `A(1, 1), B(1, 3), C(3, 3),` and `D(3, 1)`. If it is dilated with a scale factor of `1.5` centered at the point origin, find the coordinates of the vertices of the dilated square.
- `A'(1, 3), B'(5, 2), C'(2.5, 2.5),` and `D'(2.5, 1.5)`
- `A'(1.5, 1.5), B'(1.5, 4.5), C'(4.5, 4.5),` and `D'(4.5, 1.5)`
- `A'(1.5, -1.5), B'(1.5, -2.5), C'(2.5, 2.5),` and `D'(2.5, 1.5)`
- `A'(5, 3), B'(1.5, 2.5), C'(5, 5),` and `D'(2.5, 1.5)`
Answer: b
Q`3`. Figure `A` had been dilated to Figure `B`. What is the scale factor of the dilation.
- `2`
- `3`
- `2.5`
- `3.5`
Answer: c
Q`4`. The radius of a circle is `14` `\text{cm}`. What will be radius of the dilated circle if the scale factor is `0.5`?
- `12` `\text{cm}`
- `28` `\text{cm}`
- `14.5` `\text{cm}`
- `7` `\text{cm}`
Answer: d
Q5. Consider a triangle with vertices `A(1, 2), B(4, 2),` and `C(3, 4)`. Dilate the triangle with a scale factor of `2`. Write the coordinates of the vertices of the dilated triangle.
- `A'(2, 4), B'(8, 4),` and `C'(6, 8)`
- `A'(3, 4), B'(8, 4),` and `C'(2, 8)`
- `A'(2, 4), B'(1, 2),` and `C'(2, 4)`
- `A'(3, 4), B'(4, 2),` and `C'(3, 1)`
Answer: a
Frequently Asked Questions
Q`1`. What is a dilation?
Answer: Dilation is a transformation that enlarges or reduces the size of a figure without altering its shape. It involves multiplying the coordinates of each point of the figure by a constant factor, known as the scale factor.
Q`2`. What is a scale factor?
Answer: The scale factor is the ratio of the lengths of corresponding sides of the original figure and the dilated figure. It determines how much the figure is enlarged or reduced during dilation.
Q`3`. How do you perform a dilation?
Answer: To dilate a figure, you need to:
`1`. Identify the center of dilation.
`2`. Determine the scale factor.
`3`. Multiply the coordinates of each point of the original figure by the scale factor to obtain the coordinates of the dilated figure.
Q`4`. List any `2` properties of dilations?
Answer: Dilations preserve angles: The angles between intersecting lines remain unchanged after dilation.
Dilations scale lengths: Corresponding lengths on the original and dilated figures are in proportion to the scale factor.
Q`5`. What is the relationship between dilation and similarity?
Answer: Dilation is a type of similarity transformation. Similarity transformations include dilation, translation, rotation, and reflection. Dilation produces figures that are similar to the original figure, meaning they have the same shape but possibly different sizes.
Q`6`. Can a dilation have a negative scale factor?
Answer: Yes, a dilation can have a negative scale factor. A negative scale factor reflects the figure across the center of dilation. This results in a figure that is not only enlarged or reduced but also reversed or flipped.
Q`7`. How do you distinguish between enlargements and reductions in dilations?
Answer:
- Enlargement: When the scale factor is greater than `1`, the figure is enlarged.
- Reduction: When the scale factor is between `0` and `1`, the figure is reduced in size.
Q`8`. Are there any limitations or special cases in dilation geometry?
Answer:
- Division by zero: A scale factor of zero would collapse the figure to a single point, which is not practical.
- Scale factor of one: A scale factor of one would result in no change to the figure, effectively making it redundant.