Similarity as a Size Transformation: Definition, Types, Theorems

Similarity as a Size Transformation: The similarity is a property that occurs when two or more things or figures appear to be the same or equal shape. One or more rigid transformations (reflection, rotation, translation) are followed by dilation in a similarity transformation. When a figure undergoes a similarity transformation, a picture similar to the original figure is created.

In this article, we shall learn in detail about similarity as a size transformation.

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Similar Figures

Two figures (or shapes) that have the same shape and proportional dimensions but have a difference in size are said to be similar. A scaled drawing of a figure is one in which the respective lengths are proportional, and the corresponding angles are measured equally. We know that dilation results in a scaled drawing.

As a result, similar figures must be scaled drawings.

What is Similarity?

The similarity is a property that occurs when two or more things or figures appear to be the same or equal in shape. Similar figures always superimpose each other when we magnify or demagnify them.

Two circles (of any radii), for example, will always superimpose each other due to their similar property.

Congruent Figures

If polygons (or other geometric shapes) are the same size and shape, they are congruent.

Example: \(∆ABC\) and \(∆EFG\)  are congruent triangles.

Learn Criteria for Similarity of Triangles

Similarity as a Size Transformation

When two figures have the same shape but differ in size, they are said to be similar. A rigid motion together with rescaling is known as a similarity transformation. To look at it another way, a similarity transformation can change location and size while preserving the shape.

Scaled Drawing

Scaled Drawing: The shape of a scaled drawing of an object is the same as the object, but the size is different.

1. The ratio of the length of the drawing to the length of the actual thing determines the drawing’s scale.
2. A scale can be written as a size ratio or as a number-to-length ratio.
3. Consider the following scenario: \(1:400\) scale or \(1\) centimetre : \({\rm{4m}}\) scale
4. The scale is equal to the ratio of matching lengths, and the matching angles are equal.

Types of Similarity Transformation

Mainly there are four types of similarity transformations. They are given below.

1. Reflection – Mirror images are created by flipping shapes across an imaginary line.
2. Rotation – An axis is used to rotate or turn shapes.
3. Translation – Sliding shapes across the plane
4. Dilation – The process of decreasing or increasing the dimensions of an object or shape to modify its size.

Reflection

A flip is a term used in geometry to describe a reflection. The mirror image of a shape is called its reflection. The line of reflection is a line along which an image reflects. If a figure is stated to be a mirror image of another figure, then each point in the first figure is equidistant from the corresponding point in the second figure. The reflected picture should be the same shape and size as the original, but it should face the opposite direction.

Rotation

We must arrange the two triangles in the same direction, or orientation, to determine if they are identical. This is completed by rotating (turning) one shape to match the other. Rotation is the term for such a transformation.

Translation

You might think you’re looking at the same polygon twice if you’re shown a polygon and its translation. When we translate a shape, we slide it about on the plane it’s in without changing its orientation (we do not rotate it).

Dilation or Resize

The word “dilation” refers to a transformation that is used to resize an object. Dilation is a method for making items appear larger or smaller. The image produced by this transformation is identical to the original shape. However, there is a size difference in the shape. The original shape should be extended or reduced by a dilatation. The term “scale factor” defines this transformation.

Similarity Theorems

Generally, the scaling changes a triangle into another triangle with congruent angles and proportional sides similar to the original (of similarity).

Triangles’ similarity is essentially identical to their property of being congruent angles and proportional sides. There are three similarity theorems for triangles.

AAA or AA Similarity Theorem

Two triangles are similar if they have two pairs of angles that are proportionally congruent to each other.Then the two triangles are similar by AAA or AA similarity criterion.

Example:

If in \(\Delta ABC\) and \(\Delta D E F, \angle A=\angle D\) and, \(\angle B=\angle E\) then by AA criterion, \(\Delta ABC \sim \Delta DEF\)

SAS Criterion for Similarity of Triangles

Two triangles are similar if any one of their angles is equal to one of their angles, and the sides involving these angles are proportional. This criterion is known as the SAS (Side–Angle–Side) similarity criterion for two triangles.

Example:

If in \(\Delta ABC\) and \(\Delta DEF,\angle A = \angle D\) and \(\frac{A B}{D E}=\frac{A C}{D E}\) , then by SAS criterion, \(\Delta ABC \sim \Delta DEF\)

SSS Criterion for Similarity of Triangles

If three sides of one triangle are proportional to three sides with another triangle, then the triangles are similar to each other.

Consider that the sides of one triangle are proportionate to the sides of the other triangle in two triangles. Their corresponding angles are equal in that case, and the two triangles are similar. This criterion relates to the SSS (Side–Side–Side) similarity criterion for two triangles.

If in \(\Delta ABC\)  and \(\Delta PQR,\frac{{AB}}{{PQ}} = \frac{{BC}}{{QR}} = \frac{{AC}}{{PR}},\) then by “SSS” criterion, \(\Delta ABC \sim \Delta PQR\)

Testing for Similarity

If two figures have the same shape, they are said to be similar. In more mathematical terms, two figures are similar if their corresponding angles are congruent and their corresponding side length ratios are equal. The scale factor is the term given to this common ratio.

Example: \(\Delta ABC \sim \Delta DEF\)

Given triangles are \(\Delta ABC\) and \(\Delta DEF\). Here, in the given similar triangles, the corresponding sides are in proportion.

\(\frac{A B}{D E}=\frac{12}{6}, \frac{A C}{D E}=\frac{14}{7}\)

Therefore, \(\frac{A B}{D E}=\frac{A C}{D E}\) and \(\angle A=\angle D\)

So, \(\Delta ABC \sim \Delta DEF\) (By SAS similarity criteria).

Learn About Area of Similar Triangles

Solved Examples – Similarity as a Size Transformation

Q.1. In the given figure, \(\Delta PQR\) is similar to \(\Delta XYZ,\,PQ = \left( {x – 0.5} \right)\,{\rm{cm}},\,PR = 1.5\,x\,{\rm{cm}},\,XY = 9\,{\rm{cm}}\) and \(XZ = 3x\,{\rm{cm}}{\rm{.}}\) Find the lengths of \(PQ\) and \(XZ.\)

Ans: Given, \(\Delta P Q R \sim \Delta X Y Z\)
We know, in similar triangles, the corresponding sides will be in proportion.
Therefore, \(\frac{P Q}{X Y}=\frac{P R}{X Z}=\frac{Q R}{Y Z}\)
\(\Rightarrow \frac{x-0.5}{9}=\frac{Q R}{T Z}=\frac{1.5 x}{3 x}\)
\(\Longrightarrow \frac{x-0.5}{9}=\frac{1}{2}\)
\(\Rightarrow 2 x-1=9 \Longrightarrow x=5\)
Therefore, length of \(P Q=(x-0.5) \mathrm{cm}=(5-0.5) \mathrm{cm}=4.5 \mathrm{~cm}\) and 
length of \(X Z=3 x \mathrm{~cm}=3 \times 5 \mathrm{~cm}=15 \mathrm{~cm} .\)

Q.2. Are the following triangles \(\Delta ABC\) and \(\Delta DEF\) are similar? 

Ans: Given triangles are \(\Delta ABC\) and \(\Delta DEF\).
Here, in the given similar triangles, we get, \(\frac{A B}{D E}=\frac{10}{5}=2, \frac{A C}{D F}=\frac{14}{7}=2\)  and \(\frac{B C}{E F}=\frac{12}{6}=2\)
Hence, we observe that \(\frac{A B}{D E}=\frac{A C}{D F}=\frac{B C}{E F}\) This means the corresponding sides are in proportion in the two triangles.
So, \(\Delta ABC \sim \Delta DEF\)
Therefore, yes, the following triangles \(\Delta ABC\) and \(\Delta DEF\). are similar.

Q.3. Convert scale to a ratio of two numbers: \(5\,{\rm{m}}:1\,{\rm{cm}}{\rm{.}}\)
Ans: Given \(5 \mathrm{~m}: 1 \mathrm{~cm}\)
We know \(5 \mathrm{~m}=500 \mathrm{~cm}\)
\(\Rightarrow 5 \mathrm{~m}: 1 \mathrm{~cm}=500 \mathrm{~cm}: 1 \mathrm{~cm} .\)
Therefore, the obtained ratio is \(500: 1\)

Q.4. In the figure given below \(CA\) is parallel to \(BD,\,AC = 6\,{\rm{cm}},\,AE = 3\,{\rm{cm}},\,EB = 4\,{\rm{cm}},\,ED = 8\,{\rm{cm}}.\) Calculate \(CE\) and \(BD.\)

Ans: Triangles from the given figure are \(\Delta ACE\) and \(\Delta BDE\)
\(\angle A=\angle B\) (Alternate angles, since \(CA\parallel BD\)
\(\angle A C E=\angle B D E\) (Alternate angles, since \(CA\parallel BD\))
Therefore, \(\Delta ACE \sim \Delta BDE\) (AA rule of similarity)
Therefore, \(\frac{C E}{E D}=\frac{A E}{E B}=\frac{A C}{B D}\)
\(\Longrightarrow \frac{C E}{8}=\frac{3}{4}=\frac{6}{B D}\)
\(\Rightarrow C E=\frac{3}{4} \times 8=6\) and \(B D=6 \times \frac{4}{3}=8\)
Hence, \(C E=6 \mathrm{~cm}\) and \(B D=8 \mathrm{~cm} .\)

Q.5. Observe the given plane figures in which \(\angle PQR = {35^{\rm{o}}}\) and \(\angle ABC = {45^{\rm{o}}}\)  and say whether they are congruent or not.

Ans: Given \(\angle PQR = {35^ \circ }\) and \(\angle {\rm{ABC}} = {45^ \circ }\)
We know two angles are congruent if both the angles match exactly in all aspects and have equal measurements.
If we place one figure over the other figure, it will not superimpose; both figures have different measurements.
Therefore, the given figures are not congruent.

Summary

The similarity is a property that occurs when two or more things or figures appear to be the same or equal in shape. A rigid motion paired with rescaling is known as a similarity transformation. Putting it another way, a similarity transformation can change location while preserving the shape.

This article explains the definition of similar figures, congruent figures, and similarity as a size transformation and types of similarity transformation. 

This article, “similarity as a size transformations”, help in understanding these concepts in detail, and it also helps us solve the problems based on these topics very easily.

Frequently Asked Questions (FAQs)

Q.1. What is similarity transformation?
Ans: When two figures have the same shape but differ in size, they are said to be similar. A rigid motion paired with rescaling is known as a similarity transformation. To put it another way, a similarity transformation can change location while preserving the shape.

Q.2. How do you calculate similarity transformation?
Ans: If we can translate, rotate, or reflect one shape to obtain the other, the two figures are congruent. (Side lengths and angle measurements are the same.) Two figures are similar if we can translate, rotate, reflect, and/or dilate one shape to produce the other.

Q.3. What does a similarity transformation look like?
Ans: The two figures after similarity transformation look like identical figures. 

Q.4. How do you know if two figures are similar by transformations?
Ans: If two figures have the same shape, they are said to be similar. In more mathematical terms, two figures are similar if their corresponding angles are congruent and their corresponding side length ratios are equal. The scale factor is the term given to this common ratio.

Q.5. Why do we use similarity transformation?
Ans: Similarity transformations determine whether two figures have the same shape with accuracy (i.e., two figures are similar). We know that one figure is a scale representation of the other if a similarity transformation maps one figure onto another.

Q.6. How many types of similarities are there?
Ans: Mainly, there are four types of similarity transformations.
(i) Reflection
(ii) Rotation
(iii) Translation
(iv) Dilation

We hope this detailed article on the similarity as a size transformation helped you in your studies. If you have any doubts, queries or suggestions regarding this article, feel to ask us in the comment section and we will be more than happy to assist you. Happy learning!