How to find the scale factor of a dilation on a coordinate plane

Dilation means changing the size of an object without changing its shape. The size of the object may be increased or decreased based on the scale factor. An example of dilation math is a square of side 5 units can be dilated to a square of side 15 units, but the shape of the square remains the same. 

Dilation math is used to expand and contract two-dimensional or three-dimensional figures in geometry. Let us learn more about the scale factor center of dilation, and how to calculate scale factor, with the help of examples, and FAQs.

Dilation Definition

Dilation is the process of resizing or transforming an object. It is a transformation that makes the objects smaller or larger with the help of the given scale factor. The new figure obtained after dilation is called the image and the original image is called the pre-image. Dilation can be of two types:

• Expansion - When the size of an object is increased.
• Contraction - When the size of an object is decreased.

Observe the following figure which shows the dilation of a square. In this dilation, the size of the square is increased but the shape remains the same.

Center of Dilation

Dilation geometry has an important concept referred to as the 'center of dilation'. Dilation transforms the size of the figure which may either increase or decrease. The resizing happens from a point called the center of dilation. It is the center of dilation from which the objects/figures are expanded or contracted. In the figure shown below, the triangle is enlarged from the center of dilation which is marked as 'R'.

The Scale Factor

Scale factor is a number by which the size of any geometrical figure or shape can be changed with respect to its original size. It is the ratio of the sizes of the original figure with the dilated figure.

• The image is enlarged if the scale factor is more than 1 (k > 1).
• The image is contracted if the scale factor is less than 1 (0< k <1).
• The image remains the same if the scale factor is 1 (k = 1).

Note: The magnitude of the scale factor is considered and the scale factor cannot be zero.

Scale Factor Formula

The scale factor can either increase the size of an object or decrease the size of an object. The basic formula to find the scale factor of a dilated figure is:

Scale factor = Dimension of the new shape ÷ Dimension of the original shape.

This formula can be written in another way which helps to find the dimension of the new shape:  Dimensions of the original shape × Scale factor = Dimension of the new shape

Dilation in Geometry

Dilation in mathematics is a process of changing the size of an object or a shape without changing its shape. The shape can be a point, a line segment, a polygon, etc. It should be noted that the shape can be enlarged or shrunk but the proportion of each dimension of the shape and the angles remain the same. 

In the above figure,

ΔPQR is dilated (enlarged) to ΔP'Q'R' and the angles are the same. The coordinates of the vertices of ΔPQR have been changed after dilation.

P(1,3)  → P' (3,9)
Q(3,1)  → Q' (9,3)
R(1,1)  → R' (3,3)

How to Calculate the Scale Factor in Dilation?

The scale factor can be calculated when the original dimension and the changed dimension is given. Let us find the scale factor of a triangle with the original dimensions and scaled up dimensions. As seen above, after dilation, the coordinates of ΔPQR changed as follows:

P(1,3)  → P' (3,9)
Q(3,1)  → Q' (9,3)
R(1,1)  → R' (3,3)

Let us consider each vertex one by one: 

• Vertex P:
  The x-coordinate 1 changed to 3 and the y-coordinate 3 changed to 9. This shows that both the coordinates of P' became thrice the coordinates of P.
• Vertex Q:
  The x-coordinate 3 changed to 9 and the y-coordinate 1 changed to 3. This again shows that both the coordinates of Q' became thrice the coordinates of Q.
• Vertex R:
  The x-coordinate 1 changed to 3 and the y-coordinate 1 changed to 3. As we can see, both the coordinates of R' became thrice the coordinates of R

Therefore, the scale factor in this dilation is 3. Every coordinate of ΔPQR is multiplied by the scale factor of 3 to obtain the enlarged Δ P'Q'R'. In other words, if the scale factor is k, the coordinates (x,y) are transformed to (kx,ky).

(x,y)  → (kx,ky).

The scale factor can also be calculated directly by using the formula: Scale factor = Dimension of the new shape ÷ Dimension of the original shape

In this case, if we divide the coordinates of the new vertices by the coordinates of the original vertices, we can get the scale factor. For example, let us take the dimensions of vertex P (1, 3) and  P' (3, 9).

• Take the x-coordinate of P' = 3 and the x-coordinate of P = 1.
• Substitute the values in the formula: 3 ÷ 1 = 3.
• Now, take the y-coordinate of P' = 9  and the y-coordinate of P = 3.
• Apply the same formula: 9 ÷ 3 = 3. Thus, we get the scale factor of 3 from both the coordinates.

1. Example 1:  Find the scale factor of the circle with the measurements given in the figure.

   

Solution :  We see that the circle has increased in its size. Therefore, the scale factor is =  Dimension of the new shape ÷ Dimension of the original shape = Radius of the larger circle ÷  Radius of the smaller circle = 6  ÷ 3 = 2                         

Therefore, the scale factor is 2.   

Example 3: John dilated the ΔABC by a scale factor of 3 to obtain the image triangle Δ A'B'C'. The coordinates of the vertices of Δ ABC are: A(3,3), B(5,3), C(5,1). Determine the coordinates of ΔA'B'C'.

Solution:

The coordinates of the original image are A (3, 3), B (5, 3), C (5, 1) and the scale factor is 3. We know that if the scale factor is 'k', the coordinates (x,y) of any figure are transformed to (kx,ky). This can also be understood by the formula: Dimension of the new shape = Dimensions of the original shape × Scale factor

Therefore, the coordinates of the dilated (increased) image can be calculated by multiplying the original image coordinates with the given scale factor. The coordinates of Δ A'B'C' will be:

A'(3 × 3, 3 × 3) = A'(9, 9)

B'(3 × 5, 3 × 3) = B'(15, 9)

C'(3 × 5, 3 × 1) = C'(15, 3)

Therefore, A'(9, 9), B'(15, 9), C'(15, 3) are the coordinates of the dilated triangle A'B'C'.

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FAQs on Dilation Geometry

Dilation is the process of enlarging or reducing the size of a geometrical object without changing its shape. Dilation is possible by using the scale factor which helps in increasing or decreasing the size of the object.

What is a Scale Factor?

Scale factor is a number by which the size of any shape or geometrical figure can be changed with respect to its original size. A scale-up factor means the original object's size has increased and a scale-down factor means the object's size has decreased. 

What is the Center of Dilation?

The Center of Dilation is a point on a plane from which an object is expanded or contracted. 

What is Dilation Transformation?

Dilations are a kind of transformations in which an object is resized based on a scale factor. The scale factor determines how big or small a shape is. In geometry, dilation transformation means to transform the size of geometrical objects, say a square or a triangle. 

How to Find the Scale Factor of a Dilated Figure?

The scale factor can either increase or decrease the size of an object. The basic formula to find the scale factor of a dilated figure is: Scale factor = Dimension of the new shape ÷ Dimension of the original shape. For example, if a square is dilated and the side of the smaller (original) square is 4 units and the side of the dilated square (enlarged) is 12 units, then, in this case, the scale factor will be: 12 ÷ 4 = 3.