To solve the problem, let's break it down step by step.

1. Understanding the Problem:

We have two families of lines:

• $y = x + r$
• $y = -x + r$

where $r$ takes values $0, 1, 2, 3, 4$. Each family has 5 lines corresponding to these 5 values of $r$.

2. Finding the Intersection Points:

An intersection point occurs when a line from the first family intersects with a line from the second family. To find the intersection, set: $y = x + r_1 \quad \text{(from the first family)}$ equal to $y = -x + r_2 \quad \text{(from the second family)}.$

This gives: $x + r_1 = -x + r_2$ $2x = r_2 - r_1$ $x = \frac{r_2 - r_1}{2}$

Substituting $x$ back into either equation (say $y = x + r_1$), we get: $y = \frac{r_2 - r_1}{2} + r_1 = \frac{r_2 + r_1}{2}$

The point of intersection is then: $\left(\frac{r_2 - r_1}{2}, \frac{r_2 + r_1}{2}\right)$

Since both $r_1$ and $r_2$ can independently take values 0, 1, 2, 3, 4, there are $5 \times 5 = 25$ such intersection points.

3. Counting the Number of Quadrilaterals:

A quadrilateral can be formed by selecting 4 points out of the 25 intersection points. The number of ways to choose 4 points out of 25 is given by the combination formula: $\binom{25}{4} = \frac{25 \times 24 \times 23 \times 22}{4 \times 3 \times 2 \times 1} = 12650$

Thus, the number of quadrilaterals that can be formed is 12,650.

Conclusion:

The total number of quadrilaterals that can be formed by the intersection points is 12,650.

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Would you like a detailed explanation of any step or have any other questions? Here are some related problems you might find interesting:

1. How many triangles can be formed by these 25 points?
2. What if $r$ took different values, say $r$ belonged to a different set?
3. How does the problem change if only three lines from each family were considered?
4. What if we had three families of lines, say $y = ax + r$ where $a$ can also vary?
5. Can we form any special quadrilaterals (like squares or rectangles) from these points?

Tip: When working with combinatorial problems, it’s helpful to break them down into smaller, manageable parts.