The problem involves two similar quadrilaterals, ABCD and EFGH, and the task is to determine the value of $x$, which corresponds to side $AD$ in quadrilateral ABCD. Since the quadrilaterals are similar, the corresponding sides are proportional.

From the image, the side lengths are as follows:

• $AB = 48$, $BC = 24$, and $AD = x$ in quadrilateral ABCD.
• $EF = 40$, $FG = 20$, and $EH = 35$ in quadrilateral EFGH.

Using the property of similar figures, we set up a proportion between the corresponding sides:

$\frac{AB}{EF} = \frac{BC}{FG} = \frac{AD}{EH}$

Using the first pair of corresponding sides $AB$ and $EF$, and $BC$ and $FG$, we get:

$\frac{48}{40} = \frac{24}{20}$

This simplifies to:

$\frac{48}{40} = \frac{6}{5}$

Now, we can set up the proportion for $x$ (side $AD$):

$\frac{x}{35} = \frac{6}{5}$

To solve for $x$, cross-multiply:

$x \cdot 5 = 6 \cdot 35$

$5x = 210$

Now, divide by 5:

$x = \frac{210}{5} = 42$

Thus, the value of $x$ is $42$.

Would you like any further clarification? Here are some related questions you might find helpful:

1. How do we verify if two quadrilaterals are similar?
2. What other properties can we infer from similar figures?
3. Can this method be applied to three-dimensional shapes?
4. How would we approach the problem if given different side lengths?
5. What are other ways to solve problems involving proportions?

Tip: Always check if figures are similar by comparing all corresponding angles and side ratios.