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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.

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.

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.

A combination of rigid motions and dilations is called a similarity transformation. The scale factor of a similarity transformation is the product of the scale factors of the dilations involved.

Move the slider to combine rigid motions and dilations to create 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.

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.

Two figures are said to be similar if they can be mapped onto each other using only similarity transformations. Similarity between $A$ and $B$ is written as $A∼B,$ which reads $A$ is similar to $B.$

Move the slider to map one quadrilateral onto the other using similarity transformations.

The figure below is put together using $39$ similar tiles.

a How many different sizes are there on the figure?

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

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

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

a The tiles are smaller towards the bottom.

b Look for a triangular pattern.

c Move one vertex to the corresponding vertex first.

a On the figure below the different sizes are shaded using different colors.
In this pattern, five different sized tiles are used.

b 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.

c 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.

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.

- Corresponding angles are congruent.
- Corresponding sides are proportional.

A biconditional statement can be proven by showing the two properties separately.

only ifpart

If two polygons are similar, then there is a similarity transformation mapping one to the other.

- Similarity transformations preserve angle measures, so the corresponding angles of the two polygons are congruent.
- Similarity transformations map line segments to line segments. The length of all line segments change according to the scale factor, so the corresponding sides of the two polygons change proportionally.

ifpart

$∠A∠B∠C∠DABPQ =BCQR ≅∠P≅∠Q≅∠R≅∠S=CDRS =DASP $

To show similarity, a similarity transformation can be built to map $ABCD$ to $PQRS.$ This can be done in several ways, here is a possibility.

1

Translate one vertex to the corresponding vertex.

Use a translation that moves $A$ to $P.$ This translation maps $ABCD$ to $A_{′}B_{′}C_{′}D_{′}.$

2

Rotate to get the correct direction of one pair of segments.

Use a rotation around the common vertex that moves $B_{′}$ to $PQ .$ This rotation maps $A_{′}B_{′}C_{′}D_{′}$ to $A_{′′}B_{′′}C_{′′}D_{′′}.$

The following table contains some observations about the position of points $A_{′′},$ $B_{′′},$ $C_{′′},$ and $D_{′′}$ relative to the image quadrilateral.

Observation | Justification |
---|---|

$P=A_{′′}$ | The translation moves $A$ to $P$ and since this is the center of rotation, it stays there. |

$B_{′′}$ is on $PQ $ | This is how the angle of rotation was chosen. |

$D_{′′}$ is on $PS$ | This is true, because by assumption $∠A$ is congruent to $∠P$ and because rigid motions preserve angle measures. Note though that it was also used here that the orientation of $ABCD$ and $PQRS$ are the same. If the orientations are different, then a reflection in line $PQ $ is also needed to move $D_{′′}$ on $PS.$ |

3

Dilate to get a second vertex to the corresponding vertex.

Use a dilation from the common vertex to move $B_{′′}$ to $Q.$ This dilation maps $A_{′′}B_{′′}C_{′′}D_{′′}$ to $A_{′′′}B_{′′′}C_{′′′}D_{′′′}.$

The following table contains some observations about the position of points $A_{′′′},$ $B_{′′′},$ $C_{′′′},$ and $D_{′′′}$ relative to the image quadrilateral.

Observation | Justification |
---|---|

$P=A_{′′′}$ | The translation moves $A$ to $P$ and since this is the center of rotation and also the dilation, it stays there. |

$Q=B_{′′′}$ | This is how the scale factor of the dilation was chosen. |

$S=D_{′′′}$ | Since translations and rotations are rigid motions, $AB=A_{′′}B_{′′}$ and $AD=A_{′′}D_{′′}.$ It is assumed that $PQ/AB=PS/AD,$ so the dilation that moves $B_{′′}$ to $Q,$ also moves $D_{′′}$ to $S.$ |

$C_{′′′}$ is on $QR $ | It is assumed that $∠B≅∠Q.$ Since rigid motions and dilations preserve angles, this means that $∠B_{′′′}≅∠Q.$ |

$C_{′′′}$ is on $SR$ | It is assumed that $∠D≅∠S.$ Since rigid motions and dilations preserve angles, this means that $∠D_{′′′}≅∠S.$ |

$R=C_{′′′}$ | Both $R$ and $C_{′′′}$ is the intersection of $QR $ and $SR.$ |

The steps above give a similarity transformation that maps $ABCD$ to $PQRS,$ so these two quadrilaterals are similar.

Determine whether the following statements are true or false.

a All quadrilaterals are similar.

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Do all quadrilaterals have the same angles?

Different quadrilaterals may have different angles. Since similar polygons have congruent corresponding angles, this means that not all quadrilaterals are similar.

b All trapezoids are similar. {"type":"choice","form":{"alts":["True","False"],"noSort":true},"formTextBefore":"","formTextAfter":"","answer":1}

Do all trapezoids have the same angles?

Different trapezoids may have different angles. Since similar polygons have congruent corresponding angles, this means that not all trapezoids are similar.

c All parallelograms are similar. {"type":"choice","form":{"alts":["True","False"],"noSort":true},"formTextBefore":"","formTextAfter":"","answer":1}

Do all parallelograms have the same angles?

Different parallelograms may have different angles. Since similar polygons have congruent corresponding angles, this means that not all parallelograms are similar.

d All rhombi are similar. {"type":"choice","form":{"alts":["True","False"],"noSort":true},"formTextBefore":"","formTextAfter":"","answer":1}

Do all rhombi have the same angles?

Different rhombi may have different angles. Since similar polygons have congruent corresponding angles, this means that not all rhombi are similar.

e All rectangles are similar. {"type":"choice","form":{"alts":["True","False"],"noSort":true},"formTextBefore":"","formTextAfter":"","answer":1}

Consider the ratio of the sides.

The corresponding sides of similar polygons are proportional. Since $1:2 =3:4,$ the rectangles on the diagram are not similar.

Not all rectangles are similar.

f All squares are similar.

{"type":"choice","form":{"alts":["True","False"],"noSort":true},"formTextBefore":"","formTextAfter":"","answer":0}

Do all squares have the same shape?

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.

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.
### Hint

### Solution

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

$△ABC∼△ACD∼△ADE∼△AEF∼△AFG $

Find $AG.$ Give your answer rounded to the nearest millimeter. {"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":true,"useShortLog":false,"variables":[],"constants":[]}},"text":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><\/span><\/span>"},"formTextBefore":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.68333em;vertical-align:0em;\"><\/span><span class=\"mord mathdefault\">A<\/span><span class=\"mord mathdefault\">G<\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><span class=\"mrel\">=<\/span><\/span><\/span><\/span>","formTextAfter":"cm","answer":{"text":["12.2","12.20703125","\\frac{3125}{256}","12\\frac{53}{256}"]}}

Find the length of $AD$ first.

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.$CADA =BACA $

The lengths of $CA$ and $BA$ are given in the question. Substituting these values in the equation gives the length of $DA.$ $CADA =BACA $

SubstituteII

$CA=5$, $BA=4$

$5DA =45 $

MultEqn

$LHS⋅5=RHS⋅5$

$DA=45 ⋅5$

MoveRightFacToNum

$ca ⋅b=ca⋅b $

$DA=425 $

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

Proportion | Solution | |
---|---|---|

Expression | Substitution | |

$CAEA =BADA $ | $5EA =425/4 $ | $EA=425/4 ⋅5=16125 $ |

$CAFA =BAEA $ | $5FA =4125/16 $ | $FA=4125/16 ⋅5=64625 $ |

$CAGA =BAFA $ | $5GA =4625/64 $ | $GA=4625/64 ⋅5=2563125 $ |

The length of $GA$ is $2563125 ,$ or approximately $12.2$ centimeters.

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.{"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":true,"useShortLog":false,"variables":[],"constants":[]}},"text":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><\/span><\/span>"},"formTextBefore":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.48312em;vertical-align:0em;\"><\/span><span class=\"mord mathdefault\" style=\"margin-right:0.02691em;\">w<\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><span class=\"mrel\">\u2248<\/span><\/span><\/span><\/span>","formTextAfter":null,"answer":{"text":["1.47"]}}

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

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.$w_{2}x =w1 $

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.
$wy =w_{2}w ⟹y=1 $

Consider one more quadrilateral.
This time, proportionality gives an expression for $z.$
$1z =w_{2}1 ⟹z=w_{2}1 $

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=w_{2}1 +1 $

This equation is not easy to solve algebraically. Graphical calculators have applications that can solve an equation like this.
$w≈1.47 $

At the end of the solution the following equation was presented.
This is a cubic equation, the exact solution is the following.

$w=w_{2}1 +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.
$w=w_{2}1 +1$

MultEqn

$LHS⋅w_{2}=RHS⋅w_{2}$

$w_{3}=1+w_{2}$

SubEqn

$LHS−(1+w_{2})=RHS−(1+w_{2})$

$w_{3}−w_{2}−1=0$

$w=31 ⎝⎜⎛ 1+3229 −2393 +3229 +2393 ⎠⎟⎞ $

This solution can be calculated manually using Cardano's formula or it can be obtained using a computer algebra system. On the diagram all quadrilaterals are similar.

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.

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()
```

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!