Concept of Dilation
The process of dilation is the resizing of an object. It is a transformation that reduces or increases the size of things with respect to a center of dilation using a scale factor. The image is the new figure obtained after dilatation whereas the pre-image is the original figure. Depending on the scale factor, the object’s size can be increased or decreased.
The center of dilation is the point from which the objects or figures expand or contract. In the figure shown below, the triangle is enlarged from the center of dilation which is marked as ‘R’.
A scale factor tells us by how much the image has increased or decreased with respect to the figure. It’s the proportion of the dilated figure’s size to the original figure’s size.
The scale factor is usually represented by the letter k.
If the scale factor is more than one (k > 1), the image is magnified.
If the scale factor is less than one (k < 1), the image gets shrunk.
In both cases, we get a similar figure after the process of dilation.
Dilation in a Coordinate Plane
For dilating a given figure in a coordinate plane with respect to the origin, we multiply the x and y coordinates of each vertex of the figure by a scale factor, k and the origin acts as the center of dilation.
Multiple Transformations along with Dilations
First, let’s recall some transformations:
Rotation
Rotation is defined as the movement of an item around a fixed point without affecting its size or shape.
Reflection
Reflection is the process of flipping an item over a line without changing its size or shape. The dotted line is the reflection line. It’s also known as the reflection axis or the mirror line.
Translation
Translation is the process of moving a figure in any direction without affecting its size, shape or orientation.
Now let’s solve some problems that deal with various transformations including dilations.
Solved Examples on Dilation
Example 1:
ABCD are vertices of a rectangle and their coordinates are A(1, 1), B(3, 1), C(3, 2) and D(1, 2). Dilate the rectangle using a scale factor of 2. Also, translate it 6 units right and 2 units up. What are the coordinates of the image?
Solution:
This is what the graph initially looks like with the set of provided coordinates. A(1, 1), B(3, 1), C(3, 2), and D (1, 2).
The scale factor is given as 2. Hence we multiply the coordinates of the vertices by 2. The new set of points are as follows:
A’(2, 2),
B’(6, 2),
C’(6, 4),
D’(2, 4)
Translating the diagram six units to the right
A’(2+6, 2)= A’’(8,2)
B’(6+6, 2)= B’’(12,2)
C’(6+6, 4)= C’’(12,4)
D’(2+6, 4)= D’’(8,4)
Translating the diagram two units up
A’’(8,2+2)= A’’’(8,4)
B’’(12,2+2)= B’’’(12,4)
C’’(12,4+2)= C’’’(12,6)
D’’(8,4+2)= D’’’(8,6)
Example 2:
Form a reflection of the closed figure formed by connecting the points A(4, 2), B(1, 2), C(1, 0), D(4, 0) in the y-axis and then the x-axis. Dilate the same using a scale factor of 2.
These are what the following points look like when plotted along the graph A(4, 2), B(1, 2), C(1, 0) and D(4, 0)
The reflection of these points in the y-axis are as follows:
A(4, 2)= A’(-4,2)
B(1, 2)= B’(-1,2)
C(1, 0)= C’(-1,0)
D(4, 0)= D’(-4,0)
The reflection of these points in the x-axis are as follows:
A’(-4,2)= A’’(-4,-2)
B’(-1,2)= B’’(-1,-2)
C’(-1,0)= C’’(-1,0)
D’(-4,0)= D’’ (-4,0)
To dilate these points using a scale factor of 2 we multiply the coordinates by 2 and we get the following points.
A’’’(-8,-4)
B’’’(-2,-4)
C’’’(-2,0)
D’’’(-8,0)
Measure the distance between the center of dilation and a point on the preimage, as well as the distance between it and its corresponding point on the image. The ratio of these distances will also give you the scale factor
When a shape is translated, the shape moves across the coordinates but the size remains the same. When a form is mirrored, it is flipped across the coordinates but the size remains the same. When a shape is rotated, it must rotate across a point while maintaining its size. Translation, reflection and rotation are all rigid transformations, whereas dilation isn’t. On dilating, the size of the image changes