Since the triangles share the same shape, they are similar triangles. We can set up a proportion based on their corresponding sides.
For Triangle 1:
- Right side = 11.4 inches
- Bottom = 18 inches
For Triangle 2:
- Right side = \( x \) inches
- Bottom = 23.4 inches
We can set up the proportion:
\[ \frac{11.4}{18} = \frac{x}{23.4} \]
Now, we can cross-multiply to solve for \( x \):
\[ 11.4 \cdot 23.4 = 18 \cdot x \]
Calculating \( 11.4 \cdot 23.4 \):
\[ 11.4 \cdot 23.4 = 267.96 \]
So we have:
\[ 267.96 = 18x \]
Now, divide both sides by 18 to isolate \( x \):
\[ x = \frac{267.96}{18} \approx 14.88 \]
Thus, the length of \( x \) to the nearest hundredth is:
\[ \boxed{14.88} \]