Similarity - Similar Triangles and Proportions Color-By-Number Worksheet - Free Printable

Problem Overview:

The task involves solving for the variable \( x \) in each of the given similar triangles. Since all triangles are similar, we can use the property that corresponding sides of similar triangles are proportional. The goal is to solve for \( x \) in each problem and then use the solutions to color a picture according to the provided key.

Step-by-Step Solution:

#### Problem 1:
Given:
\[
\frac{3}{x} = \frac{2}{8}
\]

Solution:
Cross-multiply:
\[
3 \cdot 8 = 2 \cdot x \implies 24 = 2x \implies x = \frac{24}{2} = 12
\]

#### Problem 2:
Given:
\[
\frac{x}{20} = \frac{5}{13}
\]

Solution:
Cross-multiply:
\[
x \cdot 13 = 20 \cdot 5 \implies 13x = 100 \implies x = \frac{100}{13}
\]

#### Problem 3:
Given:
\[
\frac{5}{x} = \frac{2}{9}
\]

Solution:
Cross-multiply:
\[
5 \cdot 9 = 2 \cdot x \implies 45 = 2x \implies x = \frac{45}{2} = 22.5
\]

#### Problem 4:
Given:
\[
\frac{x}{2} = \frac{16}{4}
\]

Solution:
Simplify the right-hand side:
\[
\frac{16}{4} = 4
\]
So:
\[
\frac{x}{2} = 4 \implies x = 4 \cdot 2 = 8
\]

#### Problem 5:
Given:
\[
\frac{35}{2x} = \frac{7}{2}
\]

Solution:
Cross-multiply:
\[
35 \cdot 2 = 7 \cdot 2x \implies 70 = 14x \implies x = \frac{70}{14} = 5
\]

#### Problem 6:
Given:
\[
\frac{17}{x} = \frac{8}{3}
\]

Solution:
Cross-multiply:
\[
17 \cdot 3 = 8 \cdot x \implies 51 = 8x \implies x = \frac{51}{8}
\]

#### Problem 7:
Given:
\[
\frac{59}{x} = \frac{6}{9}
\]

Solution:
Simplify the right-hand side:
\[
\frac{6}{9} = \frac{2}{3}
\]
So:
\[
\frac{59}{x} = \frac{2}{3} \implies 59 \cdot 3 = 2 \cdot x \implies 177 = 2x \implies x = \frac{177}{2} = 88.5
\]

#### Problem 8:
Given:
\[
\frac{42}{x} = \frac{7}{5}
\]

Solution:
Cross-multiply:
\[
42 \cdot 5 = 7 \cdot x \implies 210 = 7x \implies x = \frac{210}{7} = 30
\]

#### Problem 9:
Given:
\[
\frac{47}{x} = \frac{2}{3}
\]

Solution:
Cross-multiply:
\[
47 \cdot 3 = 2 \cdot x \implies 141 = 2x \implies x = \frac{141}{2} = 70.5
\]

#### Problem 10:
Given:
\[
\frac{43}{x} = \frac{7}{5}
\]

Solution:
Cross-multiply:
\[
43 \cdot 5 = 7 \cdot x \implies 215 = 7x \implies x = \frac{215}{7}
\]

#### Problem 11:
Given:
\[
\frac{20}{x} = \frac{10}{15}
\]

Solution:
Simplify the right-hand side:
\[
\frac{10}{15} = \frac{2}{3}
\]
So:
\[
\frac{20}{x} = \frac{2}{3} \implies 20 \cdot 3 = 2 \cdot x \implies 60 = 2x \implies x = \frac{60}{2} = 30
\]

#### Problem 12:
Given:
\[
\frac{29}{x} = \frac{28}{27}
\]

Solution:
Cross-multiply:
\[
29 \cdot 27 = 28 \cdot x \implies 783 = 28x \implies x = \frac{783}{28}
\]

Final Answers:

\[
\boxed{12, \frac{100}{13}, 22.5, 8, 5, \frac{51}{8}, 88.5, 30, 70.5, \frac{215}{7}, 30, \frac{783}{28}}
\]

Coloring the Picture:

Using the provided key, match each solution to its corresponding color. For example:
- If \( x = 12 \), use the color associated with 12 in the key.
- If \( x = \frac{100}{13} \), use the color associated with \(\frac{100}{13}\) in the key, and so on.

This completes the task.