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

Conditional Statement	Two polygons are similar if the corresponding angles are congruent and the corresponding sides are proportional.
Converse	If the corresponding angles in two polygons are congruent and the corresponding sides are proportional, then the polygons are similar.

Conditional Statement

Two polygons are similar if the corresponding angles are congruent and the corresponding sides are proportional.

Converse

If the corresponding angles in two polygons are congruent and the corresponding sides are proportional, then the polygons are similar.

Observation	Justification
$P = A^{''}$	The translation moves $A$ to $P$ and, since this is the center of rotation, it stays there.
$B^{''}$ is on $P Q$	This is how the angle of rotation was chosen.
$D^{''}$ is on $P S$	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 $A B C D$ and $P Q R S$ is the same. If the orientations are different, then a reflection of $A^{''} B^{''} C^{''} D^{''}$ in line $P Q$ is also needed to match the orientations of the polygons.

Observation

Justification

P = A^{''}

The translation moves

A

P

and, since this is the center of rotation, it stays there.

B^{''}

is on

P Q

This is how the angle of rotation was chosen.

D^{''}

is on

P S

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

A B C D

and

P Q R S

is the same. If the orientations are different, then a reflection of

A^{''} B^{''} C^{''} D^{''}

in line

P Q

is also needed to match the orientations of the polygons.

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, $A B = A^{''} B^{''}$ and $A D = A^{''} D^{''} .$ It is assumed that $P Q / A B = P S / A D,$ so the dilation that moves $B^{''}$ to $Q,$ also moves $D^{''}$ to $S .$
$C^{'''}$ is on $Q R$	It is assumed that $∠ B ≅ ∠ Q .$ Since rigid motions and dilations preserve angles, this means that $∠ B^{'''} ≅ ∠ Q .$
$C^{'''}$ is on $S R$	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 $Q R$ and $S R .$

Observation

Justification

P = A^{'''}

The translation moves

A

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,

A B = A^{''} B^{''}

and

A D = A^{''} D^{''} .

It is assumed that

P Q / A B = P S / A D,

so the dilation that moves

B^{''}

Q,

also moves

D^{''}

S .

C^{'''}

is on

Q R

It is assumed that

∠ B ≅ ∠ Q .

Since rigid motions and dilations preserve angles, this means that

∠ B^{'''} ≅ ∠ Q .

C^{'''}

is on

S R

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

Q R

and

S R .

Proportion	Solution
Expression	Substitution
$\frac{E A}{C A} = \frac{D A}{B A}$	$\frac{E A}{5} = \frac{2 5 / 4}{4}$	$E A = \frac{2 5 / 4}{4} \cdot 5 = \frac{1 2 5}{1 6}$
$\frac{F A}{C A} = \frac{E A}{B A}$	$\frac{F A}{5} = \frac{1 2 5 / 1 6}{4}$	$F A = \frac{1 2 5 / 1 6}{4} \cdot 5 = \frac{6 2 5}{6 4}$
$\frac{G A}{C A} = \frac{F A}{B A}$	$\frac{G A}{5} = \frac{6 2 5 / 6 4}{4}$	$G A = \frac{6 2 5 / 6 4}{4} \cdot 5 = \frac{3 1 2 5}{2 5 6}$

Proportion

Solution

Expression

Substitution

\frac{E A}{C A} = \frac{D A}{B A}

\frac{E A}{5} = \frac{2 5 / 4}{4}

E A = \frac{2 5 / 4}{4} \cdot 5 = \frac{1 2 5}{1 6}

\frac{F A}{C A} = \frac{E A}{B A}

\frac{F A}{5} = \frac{1 2 5 / 1 6}{4}

F A = \frac{1 2 5 / 1 6}{4} \cdot 5 = \frac{6 2 5}{6 4}

\frac{G A}{C A} = \frac{F A}{B A}

\frac{G A}{5} = \frac{6 2 5 / 6 4}{4}

G A = \frac{6 2 5 / 6 4}{4} \cdot 5 = \frac{3 1 2 5}{2 5 6}

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()

Inside the larger pentagon, there is a similar smaller pentagon.

two similar pentagons where one is in the interior of the other

What is the value of

x ?

Answer in exact form.

What is the value of

y ?

Answer in exact form.

We were told that the pentagons are similar. Let's identify which sides that are corresponding. To do that, we will remove the side lengths for now and translate the smaller pentagon until it is outside of the larger pentagon.

Now, to help us identify corresponding sides, we will rotate one of the figures. Notice that there is only one acute angle in each pentagon. Use the positioning of these acute angles to manipulate the figures into the same orientation.

Great! Now, let's bring back the side lengths.

Having identified corresponding sides, we can set up a proportionality and solve for x.

To solve for y, we have to set up a proportionality that includes y and its corresponding side in the smaller pentagon.

The following map shows Long Island. What is the actual length of Long Island in miles, as measured between the blue points?

A map of Long Island with a scale of 1 in = 32 mi

In order to determine the length of Long Island, we need to know two things.

The distance between the blue dots on the map.
The map's scale.

Distance Between Blue Dots (Map)

To measure the distance between the blue dots, we must move the ruler until its starting point is on the blue dot on the left. Then we align it such that it trails the segment one can draw between the blue dots. Keep in mind that when we measure with the ruler shown, we will find the distance on the map in centimeters.

The distance between the blue dots on the map is roughly 9.7 cm.

Scale of the Map

We see the scale of the map in the lower right corner of the map.

As we can see, 1 inch on the map represents 32 miles in the real world. We can write this as the following equation. 1 inch=32 miles

Distance Between Blue Dots (Actual)

Before we calculate the actual distance, we need to convert the distance on the map into inches. We multiply it by the conversion factor 1 in2.54cm.

Since we know the distance that 1 inch on the map represents, we can determine the actual length of Long Island by multiplying both sides of the equation by 3.8.

Long Island is about 121.6 miles long. However, answers within 120 to 122 miles are accepted.

	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

Proving the Conditional Statement

Proving the 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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