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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.
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.
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.
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.
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 showing the two properties separately.
only ifpart
If two polygons are similar, then there is a similarity transformation mapping one to the other.
ifpart
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.
Use a translation that moves A to P. This translation maps ABCD to A′B′C′D′.
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. |
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.
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.
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 | |
---|---|---|
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 w2.
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.
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!