The given image shows two triangles, $\triangle FTG$ and $\triangle FSH$, which are likely similar because they share the same angles (the inner triangle $\triangle FTG$ appears to be scaled down proportionally from the larger triangle $\triangle FSH$).

Here is a step-by-step analysis and solution:

Step 1: Establish Similarity

The two triangles are similar by the Angle-Angle Similarity Theorem, as they share angle $F$ and have parallel lines, which ensure proportionality.

Step 2: Proportionality of Sides

Since the triangles are similar, the ratios of their corresponding sides are equal. Using the provided side lengths: $\frac{\text{Side in } \triangle FTG}{\text{Corresponding Side in } \triangle FSH} = \frac{\text{Other Sides in } \triangle FTG}{\text{Other Corresponding Sides in } \triangle FSH}.$

Specifically, we use: $\frac{FG}{FH} = \frac{FT}{FS} = \frac{TG}{SH}.$

Step 3: Substitute Known Values

From the diagram:

• $FG = 6x + 15$
• $FH = 88$
• $FT = 54$
• $FS = 48$

Using the proportionality: $\frac{FT}{FS} = \frac{FG}{FH}.$

Step 4: Solve for $x$

Substitute the known values: $\frac{54}{48} = \frac{6x + 15}{88}.$

Simplify $\frac{54}{48}$ to $\frac{9}{8}$: $\frac{9}{8} = \frac{6x + 15}{88}.$

Cross-multiply to eliminate fractions: $9 \times 88 = 8 \times (6x + 15).$

Simplify: $792 = 48x + 120.$

Solve for $x$: $48x = 792 - 120,$ $48x = 672.$ $x = \frac{672}{48}.$ $x = 14.$

Final Answer:

The value of $x$ is 14.

Would you like me to confirm additional calculations or explain another aspect of the problem? Here are five related questions you might consider:

1. How do we verify the similarity of triangles using angles?
2. Can we solve for the remaining side lengths once $x$ is found?
3. How does proportionality work in geometric problems?
4. What is the relationship between scale factors and side lengths?
5. How can we apply similar triangle properties to real-world scenarios?

Tip: Always simplify fractions in geometry problems early; it makes cross-multiplication much easier to handle.