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Here is some recommended reading before getting started with this lesson.
The applet below shows a spiral tiling of a plane using quadrilaterals of different sizes.
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.
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.
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.
$∼$indicates that two figures are similar.
The figure below is put together using $39$ similar tiles.
Example Answer: Translation up, followed by a rotation clockwise by $90$ degrees, followed by a dilation using scale factor $2.$
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.
A biconditional statement can be proven by separately proving the corresponding conditional statement and its converse.
Conditional Statement | Two polygons are similar if the corresponding angles are congruent and the corresponding sides are proportional. |
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Converse | If the corresponding angles in two polygons are congruent and the corresponding sides are proportional, then the polygons are similar. |
Consider and prove each statement one at a time.
If two polygons are similar, then a similarity transformation that maps one polygon to the other exists. Consider how that relationship affects the corresponding angles and sides of the similar polygons.
These observations conclude the proof of the conditional statement.
Consider two polygons with congruent corresponding angles and proportional corresponding sides. The proof here will be carried out for quadrilaterals $ABCD$ and $PQRS,$ but it can be generalized to any polygon.
Since the corresponding angles are congruent and the corresponding sides are proportional, the following statements are true.Observation | Justification |
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$P=A_{′′}$ | 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 that, in this case, the orientation of $ABCD$ and $PQRS$ is the same. If the orientations are different, then a reflection of $A_{′′}B_{′′}C_{′′}D_{′′}$ in line $PQ $ is also needed to match the orientations of the polygons. |
Observation | Justification |
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$P=A_{′′′}$ | 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 $SR.$ |
Determine whether the following statements are true or false.
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.
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.
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.
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.
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.
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 $n.$
These two properties guarantee that the squares are similar.
Any two squares are similar.
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.$CA=5$, $BA=4$
$LHS⋅5=RHS⋅5$
$ca ⋅b=ca⋅b $
The other three triangles are also similar to $△ABC,$ so the corresponding sides are proportional.
Proportion | Solution | |
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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.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.$LHS⋅w_{2}=RHS⋅w_{2}$
$LHS−(1+w_{2})=RHS−(1+w_{2})$
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.
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!