** 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,

**General steps:**

**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:**

- Each angle of the shape is the same

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.

- The dilation function of the function Y= f(x) dilated horizontally by a scale factor C is,

** Y = f(Cx)**

This kind is known as horizontal dilation. Whereas,

- The dilation function of the function Y= f(x) dilated vertically by a scale factor C is,

**Y = C * f(x) **

This kind is known as Vertical dilation.

**For example,** Consider the function y= x^{2} is dilated vertically by the scale factor 2 , then

Y = 2x^{2}

But in the case of horizontal dilation, the function will be,

Y = (2x)^{ 2}

Y= 4x^{2.}

Below are the examples on **dialation** in math

**Problem1:**

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)

**Solution:**

**Step 1:**

First we plot the triangle ABC with the given coordinates.

**Step 2:**

** **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)

**Step 3:**

Plot the above points A', B', C' and join them to get the dilated triangle.

**Problem 2:**

An image has the vertices A( 0,1), B(2,2), C(-2,2) . Perform dilation by the scale factor 2.

**Solution:**

**Step 1: **

Plot the points A( 0,1), B(2,2), C(-2,2) on the graph

**Step 2:**

Multiply each points by the dilation factor .So the new vertices are a(0,2) , b(4,4),c(-4,4)

**Step 3: **

Plot the new vertices on the graph.

Now the dilation on the graph i

**Problem3**:

Select the dilation on the graph of y = 4x^{2} to get the graph of y = 12x^{2}.

**Solution:**

By the definition of vertical dilation,

y = Ax^{2}

We can stretch the graph vertically by the scale factor of 3. Therefore we can get y =12x^{2}.

Hence the transformation we used here is vertical dilation.

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