https://file.aiquickdraw.com/mathbot/file/1731032336349hx9ngu6w.jpg

In this problem, we have two similar triangles, \( \triangle FGH \) and \( \triangle IJK \). The side lengths of these triangles are given, and we are asked to find the length of \( IJ \) (denoted as \( x \) in the diagram).

### Step-by-Step Solution
1. **Identify the corresponding sides:** Since the triangles are similar, their corresponding sides are proportional. This gives us:
   \[
   \frac{FG}{IJ} = \frac{GH}{JK}
   \]

2. **Assign known values to the ratios:**
   - \( FG = 23 \)
   - \( GH = 17 \)
   - \( IJ = x \) (unknown)
   - \( JK = 58 \)

   Using the proportion:
   \[
   \frac{23}{x} = \frac{17}{58}
   \]

3. **Solve for \( x \):**
   Cross-multiply to solve for \( x \):
   \[
   23 \times 58 = 17 \times x
   \]
   \[
   1334 = 17x
   \]
   \[
   x = \frac{1334}{17}
   \]
   \[
   x \approx 78.5
   \]

### Answer
The length of \( IJ \) is approximately **78.5**.

Would you like more details on the solution or have any questions?

Here are some related questions for further understanding:
1. How do you identify corresponding sides in similar triangles?
2. What is the cross-multiplication method, and how does it work in proportions?
3. Why are similar triangles useful in solving for unknown lengths?
4. How do you round numbers to the nearest tenth, and why is it necessary here?
5. What other real-life applications use triangle similarity and proportional reasoning?

**Tip:** Always double-check which sides correspond in similar triangles, as this ensures accurate setting up of the proportion.

Triangle FGH is similar to triangle IJK. Find IJ. Round your answer to the nearest tenth if necessary. Figures are not drawn to scale.

Learn how to solve for the missing side length IJ in similar triangles FGH and IJK using proportions. This problem involves understanding triangle similarity and applying proportional reasoning. Ideal for students in Grades 8-10.

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