To find the length of side \( x \) in the big triangle that corresponds to the side measuring 11.4 inches in the small triangle, we can use the principle of similar triangles. The sides of similar triangles are proportional.
Given:
- Side in the small triangle: 11.4 inches
- Corresponding side in the big triangle: \( x \)
- Side in the small triangle: 18 inches
- Corresponding side in the big triangle: 23.4 inches
We can set up the proportion between the sides of the triangles:
\[ \frac{11.4}{x} = \frac{18}{23.4} \]
Now, cross-multiply to solve for \( x \):
\[ 11.4 \cdot 23.4 = 18 \cdot x \]
Calculating \( 11.4 \cdot 23.4 \):
\[ 11.4 \cdot 23.4 = 267.96 \]
Now we have:
\[ 267.96 = 18x \]
To find \( x \), divide both sides by 18:
\[ x = \frac{267.96}{18} \]
Calculating \( \frac{267.96}{18} \):
\[ x \approx 14.88 \]
Thus, the length of \( x \) to the nearest hundredth is:
\[ \boxed{14.88} \]