Given that triangles \(FGH\) and \(IJH\) are similar, we can set up a relationship based on their corresponding sides.
From the information provided:
- Side \(FG\) (across the pond) is \(d\) meters.
- Side \(IJ\) is \(13\) meters.
- Segment \(FI\) is \(72\) meters.
- Segment \(IH\) is \(117\) meters.
In similar triangles, the ratios of corresponding sides are equal. This means we can write the following proportion:
\[ \frac{FG}{IJ} = \frac{FH}{IH} \]
First, we need to find another relationship involving \(FH\) (the distance from point \(F\) to point \(H\)). Since angles \(GFH\) and \(JIH\) are congruent, we can find the lengths of segments \(FH\) for both triangles.
In triangle \(FGH\):
- The right triangle can be analyzed using Pythagoras theorem where \(FH\) is the hypotenuse: \[ FH = \sqrt{FG^2 + GH^2} \]
In triangle \(IJH\):
- Similarly, \(JH\) is hypotenuse: \[ JH = \sqrt{IJ^2 + JH^2} \]
Unfortunately, the lengths for \(GH\) and \(JH\) are not explicitly given, so we can rely only on the fact that all triangles form right angles with each other and use the known side lengths of the similar triangles.
Set up the proportion:
\[ \frac{d}{13} = \frac{FH}{117} \]
To find \(FH\), we also have:
\[ FH = FI - IH = 72 - 117 = -45 \text{ (Incorrect, because lengths can't be negative)} \]
Instead, we can calculate using similar triangles: Since both triangles share a relationship via similarity in a right triangle with sides: \[ \frac{d}{13} = \frac{72}{117} \]
Cross-multiplying gives:
\[ 117d = 72 \cdot 13 \]
Calculating \(72 \cdot 13\):
\[ 72 \cdot 13 = 936 \]
Thus, we have:
\[ 117d = 936 \]
Now, solve for \(d\):
\[ d = \frac{936}{117} \]
Calculating:
\[ d = 8 \]
Therefore, the width of the pond \(d\) is:
\[ \boxed{8} \] meters.