Understanding Similarity Transformations: Scale Factor, Corresponding Sides, and Congruent Angles

Student Learning Objectives:
Identify similarity transformations and similar figures Verify similarity after similarity transformations

When working with transformations, remember that the composition of rigid motions is a rigid motion. However, since a dilation is not a rigid motion, what happens if a rigid motion and a dilation are composed? The answer to that question will be developed throughout this lesson.

Catch-Up and Review

Here is some recommended reading before getting started with this lesson.

Concept and properties of dilation.
Concept and properties of rigid motion.
Concept and properties of congruence.

Explore

Investigating Similarity

The applet below shows a spiral tiling of a plane using quadrilaterals of different sizes.

Moving the left slider dilates the shaded quadrilateral using the center.
Moving the right slider rotates the shaded quadrilateral around the center.

Move the sliders to match the quadrilaterals in the tiling.

Discussion

Similarity Transformations

On the previous applet the combination of a rotation and a dilation moved the quadrilateral to match the other quadrilaterals in the tiling. This combination of transformations has its own name.

Concept

Similarity Transformation

A similarity transformation is a combination of rigid motions and dilations. The figure it produces are similar figures. Move the slider to work through a similarity transformation step by step.

One triangle is mapped onto the other triangle using rigid motions and dilations

The scale factor of a similarity transformation is the product of the scale factors of the dilations it contains.

Discussion

Properties of a Similarity Transformation

Previously, it was seen that rigid motions keep the figure's size and shape. In comparison, dilations keep the figure's shape but can change its size. The next natural question is, what does a similarity transformation do to a figure?

The following is a list of a few important properties of similarity transformations.

The image of a line segment is a line segment. The image is longer or shorter than the preimage in the ratio given by the scale factor.
Similarity transformations preserve angle measures.

Proof

These are properties of both rigid motions and dilations. Since similarity transformations are combinations of rigid motions and dilations, these are also properties of similarity transformations.

Discussion

Definition of Similar Figures

Two figures are similar figures if there is a composition of similarity transformations that maps one figure onto the other. The symbol ~ indicates that two figures are similar.

When writing a similarity statement, the corresponding vertices are listed in the same order they appear. Two polygons can have more than one valid statement; here are two of them.

ABCD~ JKLM or CDAB~ LMJK

The same definition applies to three-dimensional shapes.

Consider one similarity statement for the given polyhedrons.

ABCDEFGH~ JKLMNOPR

Example

Analyzing Similar Figures

The figure below is put together using 39 similar tiles.

How many different sizes are there on the figure?

What is the scale factor between the smallest and largest tile?

Find a similarity transformation that maps the green tile to the blue tile. State the scale factor.

Answer

Example Answer: Translation up, followed by a rotation clockwise by 90 degrees, followed by a dilation using scale factor 2.

Hint

The tiles are smaller towards the bottom.

Look for a triangular pattern.

Move one vertex to the corresponding vertex first.

Solution

On the figure below the different sizes are shaded using different colors.

In this pattern, five different sized tiles are used.

Notice that a triangular pattern can help in finding the scale factor between the different sized tiles.

The combined width of four of the smallest tiles matches the width of the largest tile. This means that an enlargement with scale factor 4 is needed to get the largest tile from the smallest tile.

There are several ways a similarity transformation can be put together using rigid motions and dilations. One possibility is to start with a translation to move one vertex of the preimage tile to the corresponding vertex of the image tile.

Once a vertex is at the right place, a rotation can be used to position the pre-image in the right direction.

A dilation by scale factor 2 completes the transformation.

Discussion

Criteria for Similar Polygons

For polygons, similarity can be checked by considering angle measures and side lengths.

Two polygons are similar if and only if both of the following two properties hold.

The corresponding angles are congruent.
The corresponding sides are proportional.

Proof

A biconditional is prove in both directions: the conditional and its converse.

Conditional

If two polygons are similar, a similarity transformation maps one onto the other. That transformation preserves angle measures, so corresponding angles are congruent, and it scales every length by the same factor, so corresponding sides are proportional.

Converse

Suppose two polygons have congruent corresponding angles and proportional corresponding sides, as marked below for quadrilaterals ABCD and PQRS.

The goal is to build a similarity transformation that carries ABCD onto PQRS. A translation brings one vertex into place, a rotation lines up a side, and a dilation scales the figure to finish the match. Step through them in the applet.

Because these three transformations together map ABCD exactly onto PQRS, the two polygons are similar. This proves the converse and the same construction works for any polygon.

Example

Investigating Similarity of Different Polygons

Not all rectangles are similar.

All squares are similar.

Hint

Do all squares have the same shape?

Solution

Consider the angles and the sides of a square with side length m and a square with side length n.

All angles of both squares are right angles, so the corresponding angles of the two squares are congruent.
The ratio between any sides of the two squares is m:n, so the corresponding sides are proportional.

These two properties guarantee that the squares are similar.

Any two squares are similar.

Example

Solving Problems Using Similarity of Polygons

The triangles on the diagram are similar. It is given that the length of AB is 4 centimeters, the length of BC is 3 centimeters, and the length of AC is 5 centimeters. △ ABC ~ △ ACD ~ △ ADE ~ △ AEF ~ △ AFG Find AG. Give your answer rounded to the nearest millimeter.

Hint

Find the length of AD first.

Solution

Focus on the first two triangles to find the length of AD first.

It is given that △ ABC is similar to △ ACD, so the corresponding sides are proportional. DA/CA=CA/BA The lengths of CA and BA are given in the question. Substituting these values in the equation gives the length of DA.

DA/CA=CA/BA

SubstituteII

CA= 5, BA= 4

DA/5=5/4

MultEqn

LHS * 5=RHS* 5

DA=5/4* 5

MoveRightFacToNum

a/c* b = a* b/c

DA=25/4

Similar argument gives the length of EA, FA, and GA.

The other three triangles are also similar to △ ABC, so the corresponding sides are proportional.

Proportion		Solution
Expression	Substitution	Solution
EA/CA=DA/BA	EA/5=25/4/4	EA=25/4/4* 5=125/16
FA/CA=EA/BA	FA/5=125/16/4	FA=125/16/4* 5=625/64
GA/CA=FA/BA	GA/5=625/64/4	GA=625/64/4* 5=3125/256

The length of GA is 3125256, or approximately 12.2 centimeters.

Example

Solving Problems Using Similarity and Congruence of Polygons

In the diagram all quadrilaterals are similar, and the two shaded quadrilaterals are congruent. The length of three sides of the shaded quadrilaterals are 1, w, and w^2.

Find the value of w. Write your answer rounded to two decimal places.

Hint

Write the horizontal base of the top left corner's quadrilateral in two different ways.

Solution

The solution has several steps. First, look at the quadrilateral in the top left corner and compare it to the quadrilateral next to it.

These two quadrilaterals are similar, so the corresponding sides are proportional. x/w^2=1/w This gives the length of the bottom side of the quadrilateral in the top left. x=w Next, consider the quadrilaterals in the bottom left corner.

These are also similar quadrilaterals, so the corresponding sides are proportional. y/w=w/w^2 ⟹ y=1 Consider one more quadrilateral.

This time, proportionality gives an expression for z. z/1=1/w^2 ⟹ z=1/w^2 Putting these together gives the following diagram.

Comparing the two ways the length of the bottom side of the top left quadrilateral can be expressed gives an equation for w. w=1/w^2+1 This equation is not easy to solve algebraically. Graphical calculators have applications that can solve an equation like this. w≈ 1.47

Extra

Exact solution

At the end of the solution the following equation was presented. w=1/w^2+1 The approach was to use some form of technology to find the solution. It is a natural question to wonder how to find the solution algebraically. One way of doing this is to rearrange the equation.

w=1/w^2+1

MultEqn

LHS * w^2=RHS* w^2

w^3=1+w^2

SubEqn

LHS-(1+w^2)=RHS-(1+w^2)

w^3-w^2-1=0

This is a cubic equation, the exact solution is the following. w=1/3(1+sqrt(29/2-3sqrt(93)/2)+sqrt(29/2+3sqrt(93)/2)) This solution can be calculated manually using Cardano's formula or it can be obtained using a computer algebra system.

Pop Quiz

Practice Using Similarity to Solve Problems

On the diagram all quadrilaterals are similar.

Illustration

Similarity in the Natural World

There are examples where similar shapes appear in nature.

It is interesting to investigate the three-dimensional self-similar nature of a romanesco broccoli. It is built up of parts that are similar to the whole.

Self-similarity is used as an inspiration in fractals. The image below is not a living plant, it is a computer generated image using a construction that uses similarity.

Digital Tools

Python code to draw the image

The following is the Python source code the author used to draw the image.

import random
import tkinter as tk
width, height = 1024, 1024
pixels = [0] * (width * height) x, y = 0, 1
for n in range(60 * width * height): r = random.random() * 100 xn, yn = x, y if r < 1: x = 0 y = 0.16 * yn elif r < 86: x = 0.85 * xn + 0.04 * yn y = -0.04 * xn + 0.85 * yn + 1.6 elif r < 93: x = 0.20 * xn - 0.26 * yn y = 0.23 * xn + 0.22 * yn + 1.6 else: x = -0.15 * xn + 0.28 * yn y = 0.26 * xn + 0.24 * yn + 0.44 x_pix = int(width * (0.45 + 0.195 * x)) y_pix = int(height * (1 - 0.099 * y )) pixels[x_pix + y_pix * width] += 1 greys = [ max(0, (256 - p) / 256) for p in pixels]
colors = [int(c * 255) for g in greys for c in [g ** 6, g, g ** 6]]
root = tk.Tk()
p6header = bytes("P6\n{} {}\n255\n".format(width, height), "ascii")
img = tk.PhotoImage(data=p6header + bytes(colors))
tk.Label(root, image=img).pack()
img.write("barnsley-fern.png", format='png')
tk.mainloop()

Closure

Similarity and the Golden Ratio

In the diagram below move the point to adjust the dimensions of the rectangle. Different shapes will be created. Some rectangles are narrow and tall, some are wide and flat. Move the point to create rectangle of any preferred combination.

Well, personal preference is very subjective. Nevertheless, there is a specific ratio that is used often in design. If a rectangle is such that cutting off a square gives a similar rectangle, then the ratio of the sides is called the golden ratio. Such a rectangle is called a golden rectangle.

The image above is a reconstruction of Composition with Yellow, Blue and Red, a painting of Piet Mondrian, which he painted between 1937 and 1942. Move the slider points to search for golden rectangles in this famous painting!

	13 Theory slides
	11 Exercises - Grade E - A
	Each lesson is meant to take 1-2 classroom sessions

Understanding Similarity Transformations

Catch-Up and Review

Proof

Answer

Hint

Solution

Proof

Conditional

Converse

Hint

Solution

Hint

Solution

Hint

Solution

Hint

Solution

Hint

Solution

Hint

Solution

Hint

Solution

Hint

Solution

Extra

Digital Tools

Distance Between Blue Dots (Map)

Scale of the Map

Distance Between Blue Dots (Actual)

Understanding Similarity Transformations

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