Dilation is a transformation in which shape of image produced as compared to original image is same but its dimension is changed. The size of produced image can be enlarged or diminished, or in other words we can say that it describes size of an image as comared to another image. Dilation is expressed by scale factor. Scale factor is defined as it is ratio of size of newer image to older images. So we can express size of an image in terms of scale factor.
Here, triangle ABC is original triangle. After transformation image of triangle enlarged it becomes triangle A'B'C'.
We can measure size of new triangle as compared to old triangle by scale factor.if triangle A'B'C' is two times larger than triangle ABC then scale factor=size of triangle A'B'C'/size of triangle ABC
This is equal to 2*size of triangle ABC/size of triangle ABC=2
so here scale factor is 2
By another figure also we can understand concept of dilation
from the above figure we can easily find scale factor .
here coordinate of triangle abc=(2,1,2)
and coordinate of triangle a'b'c' = (4, 2, 4)
if we will multiply coordinate of triangle abc by 2 then we will get coordinates of triangle a'b'c.
so here scale factor is 2.
now from the above desription we can easily understand concept of dilation.
Steps for dilation:
The following are the steps to perform dilations in math,
Step 1: Draw the image for the given size.
Step 2: Multiply the size by the dilation factor
Step 3: Draw the new image for the new size
Steps when the vertices are given,
Step 1: Plot the given points on the graph
Step 2: Multiply each points by the dilation factor
Step 3: Now plot the new points, we get the dilated image
Step 4: Based on the dilation factor or scale factor the dilated image will be shrunken or stretched.
Properties of shapes that remain unchanged during dilation:
Parallel and Perpendicular lines in the figure remain parallel and perpendicular even after dilation.
Collinear points of the original shape remain collinear in the resultant shape obtained after dilation
Midpoints of the sides of the shapes remains the midpoints of the final shape after dilation
The image remains same and so the position of the letters is also the same
The only change is the distance between points changes.That is the length of the sides of the original shape and the image differ.
Y = f(Cx)
This kind is known as horizontal dilation. Whereas,
Y = C * f(x)
This kind is known as Vertical dilation.
For example, Consider the function y= x2 is dilated vertically by the scale factor 2 , then
Y = 2x2
But in the case of horizontal dilation, the function will be,
Y = (2x) 2
Below are the examples on dialation in math
Draw the dilated triangle of ABC about the origin with scale factor 2, where A(0,1.5) , B(-1,-1) and C(2,-1)
First we plot the triangle ABC with the given coordinates.
The scale factor is 2. Means the triangle is enlarged by a scale factor of 2. The new coordinates according to formula will be Dr(x,y) = (rx,ry)
(0,1.5) = (2*0 . 2*1.5) = A'(0,3)
(-1,-1) = (2*(-1),2*(-1)) = B'(-2,-2)
(2,-1) = (2*2,2*(-1)) = C' (4,-2)
Plot the above points A', B', C' and join them to get the dilated triangle.
An image has the vertices A( 0,1), B(2,2), C(-2,2) . Perform dilation by the scale factor 2.
Plot the points A( 0,1), B(2,2), C(-2,2) on the graph
Multiply each points by the dilation factor .So the new vertices are a(0,2) , b(4,4),c(-4,4)
Plot the new vertices on the graph.
Now the dilation on the graph i
Select the dilation on the graph of y = 4x2 to get the graph of y = 12x2.
By the definition of vertical dilation,
y = Ax2
We can stretch the graph vertically by the scale factor of 3. Therefore we can get y =12x2.
Hence the transformation we used here is vertical dilation.
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