The given image contains two parallel lines, $\overline{HI} \parallel \overline{GJ}$, and we are asked to find the length of $FJ$, a segment in the figure. The diagram shows the following distances:

• $HG = 24$
• $FG = 54$
• $IJ = 90$

Since $\overline{HI} \parallel \overline{GJ}$, the triangles $\triangle HGI$ and $\triangle GFJ$ are similar by AA (Angle-Angle) similarity, because corresponding angles formed by the parallel lines are equal.

Using the property of similar triangles:

We can write the proportion based on the similarity of the triangles: $\frac{HG}{FG} = \frac{HI}{GJ} = \frac{IJ}{FJ}$

Substitute the known values into this proportion: $\frac{24}{54} = \frac{90}{FJ}$

Solving for $FJ$:

First, simplify the fraction: $\frac{24}{54} = \frac{4}{9}$

Now set up the equation: $\frac{4}{9} = \frac{90}{FJ}$

Cross-multiply: $4 \cdot FJ = 9 \cdot 90$ $4 \cdot FJ = 810$

Now divide by 4: $FJ = \frac{810}{4} = 202.5$

Thus, the length of $FJ$ is 202.5.

Would you like a more detailed explanation or help with another question?

Here are some related questions you could explore:

1. How do you determine the similarity of two triangles geometrically?
2. What are other ways to prove that two lines are parallel in a geometric figure?
3. How does the concept of similar triangles apply to real-world scenarios?
4. Can you solve for unknown sides in a triangle using trigonometric ratios instead of similarity?
5. What happens if the lines in the figure are not parallel? How would that change the solution?

Tip: When working with similar triangles, always remember to set up the proportion correctly and check if the corresponding angles are equal.