To solve the problem, we have triangles FGH and IJH described in the prompt. Since angles FGH and IJH are right angles and angles GFH and JIH are congruent, we can conclude that triangles FGH and IJH are similar by the AA (Angle-Angle) similarity criterion.
Given the following dimensions:
- Side FG (across the pond) = d meters.
- Side IJ = 13 meters.
- Segment FI = 72 meters.
- Segment IH = 117 meters.
Let’s denote the lengths from point F to point I and from point J to point H using the lengths FI and IH as follows:
- The sides opposite the angles in right triangles:
- In triangle FGJ (denote HI as "y"):
- \[\sin \text{(angle GFH)} = \frac{FI}{FG}\]
- Thus, \[FI = 72\] forms the opposite side.
- In triangle IJH:
- \[\sin \text{(angle JIH)} = \frac{IH}{IJ}\]
- Thus, \[IH = 117\] forms the opposite side corresponding to side IJ = 13.
- In triangle FGJ (denote HI as "y"):
Now applying the property of similar triangles, we equate the ratios of corresponding sides:
\[ \frac{FI}{FG} = \frac{IH}{IJ} \]
Substituting in the known values:
\[ \frac{72}{d} = \frac{117}{13} \]
To solve for \(d\), cross multiply:
\[ 72 \times 13 = 117 \times d \]
Calculating \(72 \times 13\):
\[ 72 \times 13 = 936 \]
So, the equation becomes:
\[ 936 = 117d \]
Now, solving for \(d\):
\[ d = \frac{936}{117} \]
Calculating \(d\):
\[ d = 8 \]
Thus, the distance \(d\) across the pond is:
\[ \boxed{8} \text{ meters} \]
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