The diagram shows a triangle $DEF$ divided into two smaller triangles by segment $GH$. The given side lengths include $DE = 91$, $DG = 52$, $FH = 2x - 1$, and $GH = x + 3$. To solve for $x$, we need to determine if there is a relationship between these lengths, possibly involving similarity or proportions.

Steps to Solve:

1. Check for Triangle Similarity:

   • If triangles $\triangle DGH$ and $\triangle DEF$ are similar, their corresponding sides would be proportional.

2. Set Up Proportions:

   • Using similarity: $\frac{DG}{DE} = \frac{GH}{EF}$ Substituting the given values: $\frac{52}{91} = \frac{x + 3}{2x - 1}$

3. Cross Multiply: $52(2x - 1) = 91(x + 3)$ Expand both sides: $104x - 52 = 91x + 273$

4. Simplify the Equation: Combine like terms: $104x - 91x = 273 + 52$ $13x = 325$

5. Solve for $x$: $x = \frac{325}{13} = 25$

Final Answer:

$x = 25$

Would you like further clarification or a breakdown of any part of this solution?

Related Questions:

1. How can you confirm that triangles $\triangle DGH$ and $\triangle DEF$ are similar?
2. What happens if $x = 25$ is substituted back into $FH$ and $GH$?
3. Are there any alternative ways to verify the proportions?
4. How is cross-multiplication applied in solving proportions?
5. What other geometric properties could be used to solve this type of problem?

Tip:

When solving problems involving triangle similarity, always verify that the corresponding angles are equal or proportionality applies.