The problem states that two shapes (triangles) are similar. We are given the length of one side of the smaller triangle ($54$ in) and need to find the length of the side $h$. We are also given two corresponding side lengths of the larger triangle ($72$ in and $48$ in). We need to determine which side corresponds to the side of length $54$ in. Based on the picture, we can assume that the sides with length 54 in and 72 in are corresponding.

1. Problem Description

The problem states that two shapes (triangles) are similar. We are given the length of one side of the smaller triangle ($54$ in) and need to find the length of the side $h$. We are also given two corresponding side lengths of the larger triangle ($72$ in and $48$ in). We need to determine which side corresponds to the side of length $54$ in. Based on the picture, we can assume that the sides with length 54 in and 72 in are corresponding.

2. Solution Steps

Since the triangles are similar, the ratio of corresponding sides must be equal. We can set up a proportion to solve for $h$. We have two sets of corresponding sides: $54$ in and $72$ in, and $h$ in and $48$ in. The proportion can be set up as follows:

$\frac{h}{48} = \frac{54}{72}$

To solve for $h$, we can multiply both sides of the equation by $48$:

$h = \frac{54}{72} \times 48$

Now, we can simplify the fraction $\frac{54}{72}$ by dividing both the numerator and the denominator by their greatest common divisor, which is $18$:

$\frac{54}{72} = \frac{54 \div 18}{72 \div 18} = \frac{3}{4}$

Substitute this back into the equation for $h$:

$h = \frac{3}{4} \times 48$

$h = 3 \times \frac{48}{4}$

$h = 3 \times 12$

$h = 36$

3. Final Answer

The missing length $h$ is $36$ inches.