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Similarity, Proof, and Trigonometry

Understanding 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.

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.

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.

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 said to be similar if they can be mapped onto each other using only similarity transformations. Similarity between and is written as which reads is similar to

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

Example

Analyzing Similar Figures

The figure below is put together using similar tiles.

a How many different sizes are there on the figure?
b What is the scale factor between the smallest and largest tile?
c 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 degrees, followed by a dilation using scale factor

Hint

a The tiles are smaller towards the bottom.
b Look for a triangular pattern.
c Move one vertex to the corresponding vertex first.

Solution

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 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 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.

Proof

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

Showing the only if part

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.

Showing the if part

Consider two polygons with congruent corresponding angles and proportional corresponding sides. The proof here will be carried out for quadrilaterals and but it can be generalized to any polygon.

Two quadrilaterals, ABCD and PQRS. Corresponding angle pairs at A and P, B and Q, C and R, D and S are marked congruent.

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

1

Translate one vertex to the corresponding vertex.

Use a translation that moves to This translation maps to

2

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

Use a rotation around the common vertex that moves to This rotation maps to

The following table contains some observations about the position of points and relative to the image quadrilateral.

Observation Justification
The translation moves to and since this is the center of rotation, it stays there.
is on This is how the angle of rotation was chosen.
is on This is true, because by assumption is congruent to and because rigid motions preserve angle measures. Note though that it was also used here that the orientation of and are the same. If the orientations are different, then a reflection in line is also needed to move on


3

Dilate to get a second vertex to the corresponding vertex.

Use a dilation from the common vertex to move to This dilation maps to

The following table contains some observations about the position of points and relative to the image quadrilateral.

Observation Justification
The translation moves to and since this is the center of rotation and also the dilation, it stays there.
This is how the scale factor of the dilation was chosen.
Since translations and rotations are rigid motions, and It is assumed that so the dilation that moves to also moves to
is on It is assumed that Since rigid motions and dilations preserve angles, this means that
is on It is assumed that Since rigid motions and dilations preserve angles, this means that
Both and is the intersection of and


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

Example

Investigating Similarity of Different Polygons

Determine whether the following statements are true or false.

a All quadrilaterals are similar.

Hint

Do all quadrilaterals have the same angles?

Solution

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

Two quadrilaterals with angles 59, 76, 117, 108 and 77, 90, 62, 131 degrees each.


b All trapezoids are similar.

Hint

Do all trapezoids have the same angles?

Solution

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

Two trapezoidss with angles 76, 104, 53, 127 and 56, 124, 108, 72 degrees each.
c All parallelograms are similar.

Hint

Do all parallelograms have the same angles?

Solution

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

Two parallelograms with angles 76, 104, 76, 104 and 63, 117, 63, 117 degrees each.
d All rhombi are similar.

Hint

Do all rhombi have the same angles?

Solution

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

Two rhombi with angles 77, 103, 77, 103 and 71, 109, 71, 109 degrees each.
e All rectangles are similar.

Hint

Consider the ratio of the sides.

Solution

The corresponding sides of similar polygons are proportional. Since the rectangles on the diagram are not similar.

Two rectangles with sides 1, 3 and 2, 4 each.

Not all rectangles are similar.

f 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 and a square with side length

  • 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 so the corresponding sides are proportional.

These two properties guarantee that the squares are similar.

Two squares width side length m and n.

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 is centimeters, the length of is centimeters, and the length of is centimeters. Find Give your answer rounded to the nearest millimeter.

There are five triangles joined by their sides: ABC, ACD, ADE, AEF, and AFG.

Hint

Find the length of first.

Solution

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

Triangles ABC and ACD are highlighted.
It is given that is similar to so the corresponding sides are proportional. The lengths of and are given in the question. Substituting these values in the equation gives the length of
Similar argument gives the length of and
Triangles ABC, ADE, AEF, and AFG are highlighted.

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

Proportion Solution
Expression Substitution

The length of is or approximately 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 and

Find the value of 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. This gives the length of the bottom side of the quadrilateral in the top left. Next, consider the quadrilaterals in the bottom left corner.

These are also similar quadrilaterals, so the corresponding sides are proportional. Consider one more quadrilateral.

This time, proportionality gives an expression for 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 This equation is not easy to solve algebraically. Graphical calculators have applications that can solve an equation like this.

Extra

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

Cross section of a nautilus <credit></credit>

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.

Romanesco broccoli <credit></credit>

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.

Barnsley fern <credit></credit>

Digital Tools

Python code to draw the image

unfold_more

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()
info

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 and Move the slider points to search for golden rectangles in this famous painting!
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