To solve this problem, we need to determine the coordinates of the rotation point $P(r, s)$ such that the line segment from $A(1, 2)$ to $B(3, 3)$ is rotated to the line segment from $A'(3, 1)$ to $B'(4, 3)$.

Step 1: Analyze the Coordinates Before and After Rotation

1. The original segment has points $A(1, 2)$ and $B(3, 3)$.
2. After rotation, the points are $A'(3, 1)$ and $B'(4, 3)$.

Step 2: Find the Midpoints of $AB$ and $A'B'$

Since the rotation keeps the distance between points constant, the rotation point $P(r, s)$ should lie on the midpoint of the line joining the midpoints of $AB$ and $A'B'$.

1. Midpoint of $AB$: $\left( \frac{1 + 3}{2}, \frac{2 + 3}{2} \right) = \left( 2, \frac{5}{2} \right)$

2. Midpoint of $A'B'$: $\left( \frac{3 + 4}{2}, \frac{1 + 3}{2} \right) = \left( \frac{7}{2}, 2 \right)$

Step 3: Calculate the Midpoint Between These Two Midpoints

The rotation point $P(r, s)$ should be the midpoint of $\left( 2, \frac{5}{2} \right)$ and $\left( \frac{7}{2}, 2 \right)$.

1. Calculating $r$: $r = \frac{2 + \frac{7}{2}}{2} = \frac{4 + 7}{4} = \frac{11}{4}$

2. Calculating $s$: $s = \frac{\frac{5}{2} + 2}{2} = \frac{5 + 4}{4} = \frac{9}{4}$

Thus, $P(r, s) = \left( \frac{11}{4}, \frac{9}{4} \right)$.

Step 4: Find $|r - s|$

Now, we compute $|r - s|$: $|r - s| = \left| \frac{11}{4} - \frac{9}{4} \right| = \left| \frac{2}{4} \right| = \frac{1}{2}$

Final Answer

$\boxed{\frac{1}{2}}$

Would you like further clarification, or do you have any questions?

Related Questions:

1. What is the formula for finding the midpoint of two points?
2. How do you determine the center of rotation given two line segments?
3. What are some common properties of rotations in coordinate geometry?
4. Can the distance between points help verify rotation transformations?
5. How does the distance formula play a role in finding the midpoint or rotation center?

Tip: In rotation problems, the midpoint technique is useful for finding a center point between shifted line segments.