The problem involves similar polygons, where the corresponding sides are proportional. Here's how to solve it step-by-step:

Step 1: Identify the corresponding sides

From the diagram:

• $x+1$ in polygon $WXYZ$ corresponds to $8$ in polygon $TSRP$.
• $3x+1$ in polygon $WXYZ$ corresponds to $20$ in polygon $TSRP$.

Since the polygons are similar, the ratios of corresponding sides are equal: $\frac{x+1}{8} = \frac{3x+1}{20}$

Step 2: Solve the proportion

Cross-multiply to eliminate the fractions: $20(x+1) = 8(3x+1)$

Expand both sides: $20x + 20 = 24x + 8$

Simplify: $20 = 4x + 8$

Subtract 8 from both sides: $12 = 4x$

Divide by 4: $x = 3$

Final Answer:

$x = 3$

Would you like to see a detailed explanation of any step?

Relative Questions:

1. What are the conditions for two polygons to be similar?
2. How do you solve equations involving proportions in geometry?
3. Can we confirm that the solution satisfies all given ratios?
4. What would happen if $x$ had no solution—what would that indicate about the polygons?
5. How can we use similar polygons in practical applications like map scaling?

Tip:

When solving proportions involving variables, always double-check by substituting the solution back into the original ratios.