Dilation Geometry

Dilation a transformative process in geometry involves resizing a figure while maintaining its shape around a fixed center point. The essence of dilation lies in its ability to expand or contract geometric shapes providing a versatile tool for mathematical analysis. Dilation in Geometry is a geometric operation that introduces the concept of a scale factor determining the degree of enlargement or reduction applied to a figure.

Exploring dilation helps in establishing similarity between figures which is a key aspect in geometry and real-world applications. When a figure undergoes dilation, all its points move radially from or towards the fixed center emphasizing the directional nature of this transformation. In practical terms, dilation finds relevance in fields like cartography where maps undergo scaling to represent different geographical areas accurately.

Dilation Meaning in Math

Dilation in mathematics refers to the transformation of a figure by resizing it, either enlarging or reducing, while maintaining its shape. It is a fundamental concept in geometry and is often characterized by a scale factor. Dilation in geometry involves the transformation of a figure by resizing it concerning a fixed center point. This geometric operation preserves shape but alters size. It is a fundamental concept in geometry, commonly described by a scale factor that determines the degree of enlargement or reduction.

Dilation

Dilation finds applications in various mathematical and real-world contexts such as mapping, image processing and similarity studies. Understanding the properties and applications of dilation contributes to geometric transformations.

Dilation in Geometry

In geometry, dilation plays an important role in transforming shapes while preserving their geometric properties. It is employed to study and analyze various figures contributing to a deeper understanding of spatial relationships. The image added shows the dilation of triangle, here a triangle ABC is dilated to form a triangle DEF.

Dilation in Triangle

Center of Dilation

The center of dilation is the fixed point about which the figure undergoes its size transformation. In dilations, every point on the figure moves outward or inward from this center, maintaining a proportional relationship. In the first image of dilation added above R is the centre of Dilation

Dilation with Scale Factor

The scale factor determines the amount of enlargement or reduction in a dilation. It is a ratio comparing the corresponding lengths of the pre-image and image. A scale factor greater than 1 indicates enlargement, while a factor between 0 and 1 implies reduction.

How to Calculate Scale Factor in Dilation?

To calculate the scale factor in dilation, divide the length of a corresponding side in the image by its corresponding side in the pre-image. This provides insight into the proportional relationship and helps understand the extent of the dilation.

Below are the steps to Calculate the Scale Factor in Dilation:

• The scale factor (h) can be found by comparing corresponding lengths in the original and dilated figures.

h = {length in dilated figure}/ length in dilated figure

Example: If AB in the original triangle is 6 units and AB’ in the dilated triangle is 12 units find the scale factor.

Solution:

Given,

• Original Height = 6
• Dilated Height = 12

h = Dilated Height/Original Height = 12/6 = 2

Scale Factor Formula

When performing a dilation with a scale factor represented by h

Scale Factor = Dimension of New Shape / Dimension of Original Shape

For coordinates (x, y) and scale factor h the formula for dilated coordinates are,

Formula for Dilated Coordinates: (x’, y’) = (hx, hy)

Example: If the original point is (3, 4) and the scale factor h = 2 find the dilated point.

Solution:

Given,

• Point = (3, 4)
• Scale Factor(h) = 2

Dilated Point = {(2)(3), (2)(4)} = (6, 8)

Properties of Dilation

Dilation preserves angles but alters distances. Shapes remain similar, and corresponding sides maintain a proportional relationship. The center of dilation serves as the anchor point for this transformation.

Below listed are the Properties of Dilation

• Orientation of the object preserved.
• Dilation changes the size of the object however maintaining similarity.
• Invariant under translation, rotation and reflection.

Horizontal Dilation and Vertical Dilation

Horizontal Dilation: Multiplies the x-coordinates by the scale factor.

(x’, y’) = (hx, y)

Vertical Dilation: Multiplies the y-coordinates by the scale factor.

(x’, y’) = (x, Vy)

Example: Dilation of Point (2, 5) with horizontal scale factor 4

Solution:

Horizontal Dilation: (2.4, 5) = (8, 5)

Read More,

• Triangles in Geometry
• Rhombus
• Kite – Quadrilaterals
• Types of Angles

Solved Examples on Dilation Geometry

Example 1: Given an Original Triangle A(0,0), B(4,0), C(2,3) and a scale factor of k = 2 find the co-ordinates of the dilated triangle.

Solution:

Step 1: Define the Scale Factor (k)

In this case, k = 2

Step 2: Write the Dilation Formula

For a point (x, y)(x,y), the coordinates after dilation (x’, y’) are given by

• x’ = k⋅x
• y’ =k⋅y

Step 3: Apply Dilation to Each Point:

For point A(0, 0)

• x’ = 2⋅0 =0
• y’ = 2⋅0=0

So, A’ is still at (0, 0)

For point B(4, 0)

• x’ = 2⋅4=8
• y’ = 2⋅0=0

B’ is now at (8, 0)

For point C (2, 3)

• x’ = 2⋅2 = 4
• y’ = 2⋅3 = 6

C’ is now at (4,6)

Dilated Triangle: A'(0, 0), B'(8, 0), C'(4, 6)

Example 2: Given an Original Square: A(1,1), B(1,4), C(4,4), D(4,1) with a scale factor k = 0.5, find the Dilated Square co-ordiantes.

Solution:

Step 1: Define the Scale Factor (k):

In this case,

k = 0.5

Step 2: Write the Dilation Formula:

For a point (x, y), the coordinates after dilation

(x’, y’) = k⋅x

y’ = k⋅y

Step 3: Apply Dilation to Each Point:

For point A(1, 1):

• x’ = 0.5⋅1 = 0.5
• y’ = 0.5⋅1 = 0.5

So, A’ is now at (0.5, 0.5)

For point B(1,4):

• x’ = 0.5⋅1 = 0.5
• y’ = 0.5⋅4 = 2

B’ is now at (0.5, 2)

For point C(4,4):

• x’ = 0.5⋅4 = 2
• y’ = 0.5⋅4 = 2

C’ is now at (2, 2).

For point D(4,1):

• x’ = 0.5⋅4 = 2
• y’ = 0.5⋅1 = 0.5

D’ is now at (2, 0.5)

Dilated Square: A'(0.5, 0.5), B'(0.5, 2), C'(2, 2), D'(2, 0.5)

Practice Questions on Dilation Geometry

Q1. What happens to angles during dilation?

Q2. If the scale factor is 2, how does the image compare to the pre-image?

Q3. Identify the center of dilation in a given geometric transformation.

FAQs on Dilation in Geometry

1. What is Dilation in Geometry?

Dilation in geometry is a transformation that involves resizing a figure around a fixed center point while preserving its shape.

2. What is a Scale Factor?

The scale factor is the ratio of corresponding lengths of the image and pre-image in a dilation, determining the degree of enlargement or reduction.

3. What Is Center of Dilation?

The center of dilation is the fixed point about which a figure undergoes size transformation in dilation.

4. What is an Example of a Dilation?

Enlarging or reducing a rectangle while keeping its angles unchanged is an example of dilation.

5. What is Dilation of Size?

The dilation of size refers to the scale factor applied to transform a figure in dilation, influencing its size change.

6. How is Dilation Different From Other Geometric Transformations?

Dilation differs from translation, rotation and reflection as it focuses on changing the size of a figure while keeping its shape constant.

7. What is Real-World Applications of Dilation?

Dilation finds applications in fields like cartography, design, medical imaging and computer graphics where precise scaling is important.