Dilation
The term “dilation” refers to a transformation that is used to resize an object. Dilation is a technique for making objects appear larger or smaller.
After the dilation process, the triangle ACB is transformed into a larger triangle, A’B’C’, as shown in the diagram above. As a result, it’s a case of enlarging the size of an object or shape.
What is the Scale Factor?
The scale factor refers to the size by which a shape is enlarged or reduced. We use it when we need to make a shape larger, such as a circle, a triangle, a square, a rectangle, or any other shape.
What is a Similarity Transformation?
Dilations aren’t rigid motions because they don’t preserve length. A dilation or a sequence of dilations and rigid motions is a similarity transformation.
In the figure above, \( \Delta DEF \) is a dilation of \( \Delta ABC \) using a constant scale factor. Hence, \( \Delta ABC\sim \Delta DEF \), “\( \sim \) “ denotes that \( \Delta ABC \) is similar to \( \Delta DEF \).
What are Similar Figures?
Definition: When a figure has gone through a similarity transformation, the new figure and the original figure are similar. That is, when they have the same shape but differ in size, they are said to be similar.
Properties of Similar Figures
Similar figures, as we all know, are basically two different sizes of the same figure. The corresponding sides of similar figures are in proportion and the corresponding angles are congruent.
How can we Identify Similar Figures?
If the corresponding angles are all congruent and the corresponding sides are proportional, the shapes are similar.
Take a look at the angles and side lengths of the trapezoids below, for example.
The angles of the trapezoids are congruent because they are similar.
\( \angle E=\angle W=135^{\circ} \)
\( \angle F=\angle X=90^{\circ} \)
\( \angle G=\angle Y=90^{\circ} \)
\( \angle H=\angle Z=45^{\circ} \)
The sides of the trapezoids are proportional to one another. As a result, each pair of corresponding sides has the same ratio.
\( \frac{WX}{EF}=\frac{18}{6}=3 \)
\( \frac{XY}{FG}=\frac{12}{4}=3 \)
\( \frac{YZ}{GH}=\frac{33}{11}=3 \)
\( \frac{ZW}{HE}=\frac{15}{5}=3 \)
This ratio is the scale factor, which is 3. That is to say, the side lengths of WXYZ are three times those of EFGH.
How do we Describe a Similarity Transformation?
Let’s take an example and learn how to describe the similarity transformation.
Example: Explain how to map the trapezoid PQRS to the trapezoid WXYZ using a similarity transformation.
Solution:
- We can see that the orientation of the trapezoid PQRS to the trapezoid WXYZ is not the same. If the trapezoid PQRS is reflected in the y-axis as shown, the image trapezoid P’ Q’ R’ S’ has the same orientation as the trapezoid WXYZ.
- WXYZ appears to be one-third the size of the trapezoid P’ Q’ R’ S’. Using a scale factor of \( \frac{1}{3} \), dilate the trapezoid P’ Q’ R’ S’ with respect to the origin.
As a result, a similarity transformation that maps the trapezoid PQRS to the trapezoid WXYZ is a negative axis reflection followed by a 1/3 scale factor dilation.
Solved Similar Figures Examples
Example 1:
Examine the similarities between \( \Delta ABC \) and \( \Delta KLM \).
Solution:
The coordinates of the vertices should be compared.
A(0,2)⟶A‘(2⋅0,2⋅2)⟶K(0,4)
B(2,-1)⟶B‘(2⋅2,2⋅(-1))⟶L(4,-2)
C(-2,-2)⟶C‘(2⋅(-2),2⋅(-2))⟶M(-4,-4)
\( \Delta KLM \) is a dilation of \( \Delta ABC \) using a scale factor of 2.
So, \( \Delta ABC \) and \( \Delta KLM \) are similar.
Example 2:
\( \Delta STU \) and \( \Delta PQR \) are similar. Describe the transformation of the figures’ similarity.
Solution:
You can see that the \( \Delta S’T’U’ \) is half the size of the \( \Delta PQR \) by comparing side lengths. So, using a scale factor of \( \frac{1}{2} \), dilate the \( \Delta PQR \) with respect to the origin.
Now we can see that for \( \Delta S’T’U’ \) to be identical to \( \Delta PQR \), it has to be flipped. You must reflect the \( \Delta S’T’U’ \) in the –axis after it has been dilated.
So, using a scale factor of \( \frac{1}{2} \), dilating the \( \Delta PQR \) with respect to the origin, and then reflecting the image in the y-axis, is one possible similarity transformation.
Example 3:
Find the height (h) of the tree.
Solution:
The side lengths of the triangles are proportional because they are similar. As a result, the ratios 84 : 12 and h : 2 are the same, as the scale factor is constant.
\( \frac{84}{12}=\frac{h}{2} \) write the proportion
\( 2\times \frac{84}{12}=2\times \frac{h}{2} \) The multiplication property of equality
\( 14=h \) Simplify
So, the height of the tree is 14 meters.
The congruence and similarity of figures are distinguished by the fact that similar shapes can be resized versions of the same shape, whereas congruent figures have the exact same lengths.
Different circles can have different radii, and hence different sizes. However, the shape of all circles is the same, that is, they are all round.
A similar figure is obtained by a similarity transformation. So, when a figure is dilated, the sides of the figure are changed by a constant which is the scale factor. The angles remain the same. There is another way to understand this. Say you have an equilateral triangle that measures 5 cm. It is dilated by a scale factor of 2, to form another equilateral triangle, where each side measures 10 cm. But the angles of the equilateral triangle remain the same, which is 60 degrees. Hence, in a similarity transformation, only the sides change, and the angles remain the same.