In the context of geometry, “dilation is a substantial change in the image size without changing its original form”.

What is Dilation?

In this picture, what do you observe specifically? Do not be concerned! Let us tell you about it! In the picture above:

We have triangle ABC within the coordinate plane.
When we dilate, we get a new triangle A'B'C', which is the larger size of the original triangle.
Notice how the shape of the triangle doesn't change.
The dilation in distance of the coordinates can be checked by using our free online dilation calculator.

Types of Dilation:

1. Enlargement

When a image grows larger immediately after the dilation process, then it is termed to as enlargement.

2. Reduction

If there is some contraction happens in the image size after you carry out dilation, then it is a reduction.

3. Horizontal Dilation

This type of dilation for a function y = f(x), which dilates the function by a scale factor C by the equation y = f(Cx)

4. Vertical Dilation

The function y = f(x) dilates vertically by a scale factor C by the equation below: y = C.f(x)

What Is Scale Factor in Dilation?

“The ratio of the size of the modified image to the original one is known as the scale factor.”

The scale factor defines the change in image size. Keep in mind this:

If SF > 1, the image spreads.

If 0 < SF < 1, the image will reduce.

If SF = 1, the image remains unaltered.

Characteristics of Dilation:

After dilation, the only change is the distance between the coordinates. After dilation, the only change is the distance between the coordinates. If you want to measure the length of a vector between points, you can use the Vector Magnitude Calculator. You will also note that several aspects remain intact while the images are transformed from their original to adjusted sizes. These consist of:

Every detail in the figure remains the same.
Lines that are perpendicular and parallel also remains the same.
Both the original and dilated images keep the uniform side midpoints.

How do you calculate the Scale Factor of a Dilation?

You must be thinking that it is going to be something tricky. But believe us, it is not. Let us solve some examples so that you understand dilation properly. Stay focused!

Example 1: Enlargement

Suppose we have a triangle ABC with the following coordinates:

A(2, 3)
B(4, 5)
C(6, 7)

We perform a dilation with a scale factor of 2.

Solution:

Multiply each coordinate by the scale factor (2):

If you are working with coordinate vectors, you can also combine them using the Vector Addition Calculator.

A'(2 × 2, 3 × 2) = A'(4, 6)
B'(4 × 2, 5 × 2) = B'(8, 10)
C'(6 × 2, 7 × 2) = C'(12, 14)

Thus, the new coordinates of the dilated triangle are:

A'(4, 6)
B'(8, 10)
C'(12, 14)

Result:

The triangle has enlarged with a scale factor of 2, doubling its size while maintaining the same shape.

Example 2: Reduction

Now, consider a rectangle with the following coordinates:

P(2, 5)
Q(7, 3)
R(9, 1)
S(4, 6)

Now, let's apply a scale factor of 0.8 (a reduction).

Dilation with a scale factor of 0.8:

Multiply each coordinate by the scale factor (0.8):

P'(2 × 0.8, 5 × 0.8) = P'(1.6, 4)
Q'(7 × 0.8, 3 × 0.8) = Q'(5.6, 2.4)
R'(9 × 0.8, 1 × 0.8) = R'(7.2, 0.8)
S'(4 × 0.8, 6 × 0.8) = S'(3.2, 4.8)

Thus, the new coordinates of the scaled rectangle are:

P'(1.6, 4)
Q'(5.6, 2.4)
R'(7.2, 0.8)
S'(3.2, 4.8)

Result:

The rectangle has been reduced by a scale factor of 0.5, shrinking its size while keeping the shape intact.

Example 3: Enlargement with a Larger Scale Factor

Let's take another triangle XYZ, with the following coordinates:

X(2, 3)
Y(4, 5)
Z(6, 7)

Now, let's perform a dilation with a scale factor of 0.5.

Dilation with a scale factor of 0.5:

Multiply each coordinate by the scale factor (0.5):

X'(2 × 0.5, 3 × 0.5) = X'(1, 1.5)
Y'(4 × 0.5, 5 × 0.5) = Y'(2, 2.5)
Z'(6 × 0.5, 7 × 0.5) = Z'(3, 3.5)

Thus, the new coordinates of the dilated triangle are:

X'(1, 1.5)
Y'(2, 2.5)
Z'(3, 3.5)

Result:

The triangle has been enlarged by a scale factor of 3, making it three times the size of the original while keeping its original shape.

This content guide explains dilation, scale factor, enlargement, reduction, and examples of dilation in coordinate geometry. For an instant calculation, use our online dilation calculator to get accurate results within a second!

Dilation Calculator