To find the length of the interval $AB$ between points $A$ and $B$ on the coordinate plane, we can use the distance formula:

$AB = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$

From the image:

• Point $A$ has coordinates $(2, 2)$.
• Point $B$ has coordinates $(6, 5)$.

Substitute these values into the formula: $AB = \sqrt{(6 - 2)^2 + (5 - 2)^2}$

Calculate step-by-step:

1. $6 - 2 = 4$ and $5 - 2 = 3$.
2. Square these results: $4^2 = 16$ and $3^2 = 9$.
3. Add the squares: $16 + 9 = 25$.
4. Take the square root: $\sqrt{25} = 5$.

So, the length of the interval $AB$ is: $AB = 5$

Would you like further details on this solution or have any other questions?

Here are five related questions to deepen understanding:

1. What is the distance formula, and why does it work for finding the length between two points?
2. How would the length of $AB$ change if point $B$ were at $(6, 6)$?
3. What is the midpoint of the line segment $AB$ with points $A$ at $(2, 2)$ and $B$ at $(6, 5)$?
4. How can the distance formula be derived from the Pythagorean theorem?
5. How would you calculate the distance between two points in three-dimensional space?

Tip: Always double-check the coordinates before substituting them into the distance formula to avoid simple errors.