To find the distance between two points $A(1,1)$ and $B(-2,7)$ in a coordinate plane, we use the distance formula:

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

Here, $(x_1, y_1)$ are the coordinates of point A and $(x_2, y_2)$ are the coordinates of point B. Substituting the given values:

$(x_1, y_1) = (1, 1)$ $(x_2, y_2) = (-2, 7)$

Now, plug these into the distance formula:

$\text{Distance} = \sqrt{((-2) - 1)^2 + (7 - 1)^2}$ $\text{Distance} = \sqrt{(-3)^2 + 6^2}$ $\text{Distance} = \sqrt{9 + 36}$ $\text{Distance} = \sqrt{45}$ $\text{Distance} = 3\sqrt{5}$

So, the distance between points A and B is $3\sqrt{5}$.

Do you want any further details or have any questions?

Here are some related questions you might find interesting:

1. How do you find the midpoint of a line segment between two points?
2. What is the slope of the line passing through two given points?
3. How do you derive the equation of a line given two points?
4. How can you determine if three points are collinear?
5. How do you find the area of a triangle given its vertices?

Tip: Always double-check your calculations to avoid simple arithmetic errors, especially when working with square roots and distances.