Answer :
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]
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]