Two figures are similar if they have the same shape. Two or more figures are similar if the corresponding angles are equal, and the corresponding sides are in proportion.

The following triangles are similar. What is the value of the unknown side?

Solution:

1. Find the corresponding sides and write a proportion:
\[
\frac{4}{12} = \frac{x}{9}
\]
Now, use the cross product to solve for \( x \):
\[
\frac{4}{12} = \frac{x}{9} \rightarrow 4 \times 9 = 12 \times x \rightarrow 36 = 12x
\]
Divide both sides by 12:
\[
12x = 36 \rightarrow \frac{36}{12} = \frac{12x}{12} \rightarrow x = 3
\]
The missing side is 3.

2.
\[
\frac{27}{33} = \frac{9}{x} \rightarrow 27x = 9 \times 33 \rightarrow 27x = 297 \rightarrow x = \frac{297}{27} = 11
\]

3.
\[
\frac{10}{5} = \frac{8}{x} \rightarrow 10x = 5 \times 8 \rightarrow 10x = 40 \rightarrow x = \frac{40}{10} = 4
\]

4.
\[
\frac{6}{x} = \frac{18}{24} \rightarrow 6 \times 24 = 18 \times x \rightarrow 144 = 18x \rightarrow x = \frac{144}{18} = 8
\]

5.
\[
\frac{32}{40} = \frac{8}{x} \rightarrow 32x = 8 \times 40 \rightarrow 32x = 320 \rightarrow x = \frac{320}{32} = 10
\]

6.
\[
\frac{30}{45} = \frac{x}{6} \rightarrow 30 \times 6 = 45 \times x \rightarrow 180 = 45x \rightarrow x = \frac{180}{45} = 4
\]

Let's carefully solve each proportion step-by-step to find the unknown side. The subject is similar triangles, where corresponding sides are proportional.

1. First proportion:
[tex]\[
\frac{4}{12} = \frac{x}{9}
\][/tex]
Cross-multiplying:
[tex]\[
4 \times 9 = 12 \times x \implies 36 = 12x
\][/tex]
Solving for [tex]\(x\)[/tex]:
[tex]\[
x = \frac{36}{12} \implies x = 3
\][/tex]
The missing side is [tex]\(x = 3\)[/tex].

2. Second proportion:
[tex]\[
\frac{27}{33} = \frac{9}{x}
\][/tex]
Cross-multiplying:
[tex]\[
27 \times x = 33 \times 9 \implies 27x = 297
\][/tex]
Solving for [tex]\(x\)[/tex]:
[tex]\[
x = \frac{297}{27} \implies x = 11
\][/tex]
The missing side is [tex]\(x = 11\)[/tex].

3. Third proportion:
[tex]\[
\frac{10}{5} = \frac{8}{x}
\][/tex]
Cross-multiplying:
[tex]\[
10 \times x = 5 \times 8 \implies 10x = 40
\][/tex]
Solving for [tex]\(x\)[/tex]:
[tex]\[
x = \frac{40}{10} \implies x = 4
\][/tex]
The missing side is [tex]\(x = 4\)[/tex].

4. Fourth proportion:
[tex]\[
\frac{6}{x} = \frac{18}{24}
\][/tex]
Cross-multiplying:
[tex]\[
6 \times 24 = 18 \times x \implies 144 = 18x
\][/tex]
Solving for [tex]\(x\)[/tex]:
[tex]\[
x = \frac{144}{18} \implies x = 8
\][/tex]
The missing side is [tex]\(x = 8\)[/tex].

5. Fifth proportion:
[tex]\[
\frac{32}{40} = \frac{8}{x}
\][/tex]
Cross-multiplying:
[tex]\[
32 \times x = 40 \times 8 \implies 32x = 320
\][/tex]
Solving for [tex]\(x\)[/tex]:
[tex]\[
x = \frac{320}{32} \implies x = 10
\][/tex]
The missing side is [tex]\(x = 10\)[/tex].

6. Sixth proportion:
[tex]\[
\frac{30}{45} = \frac{x}{6}
\][/tex]
Cross-multiplying:
[tex]\[
30 \times 6 = 45 \times x \implies 180 = 45x
\][/tex]
Solving for [tex]\(x\)[/tex]:
[tex]\[
x = \frac{180}{45} \implies x = 4
\][/tex]
The missing side is [tex]\(x = 4\)[/tex].

So, summarizing the missing sides:
1. [tex]\(x = 3\)[/tex]
2. [tex]\(x = 11\)[/tex]
3. [tex]\(x = 4\)[/tex]
4. [tex]\(x = 8\)[/tex]
5. [tex]\(x = 10\)[/tex]
6. [tex]\(x = 4\)[/tex]