Find the distance between the two points in simplest radical form. (1,1) (-8,-3)

To find the distance between two points \((x_1, y_1)\) and \((x_2, y_2)\) in a coordinate plane, we use the distance formula:

\[
d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\]

Given the points \((1, 1)\) and \((-8, -3)\):

1. Assign the coordinates:
   - \(x_1 = 1\), \(y_1 = 1\)
   - \(x_2 = -8\), \(y_2 = -3\)

2. Substitute these values into the distance formula:

\[
d = \sqrt{(-8 - 1)^2 + (-3 - 1)^2}
\]

3. Simplify the expression:

\[
d = \sqrt{(-9)^2 + (-4)^2}
\]

\[
d = \sqrt{81 + 16}
\]

\[
d = \sqrt{97}
\]

So, the distance between the points \((1, 1)\) and \((-8, -3)\) is \(\sqrt{97}\).

Would you like more details or have any questions? Here are some related questions:

1. How do you find the midpoint between two points?
2. What is the equation of the line passing through the points \((1, 1)\) and \((-8, -3)\)?
3. How do you determine if two points are collinear with a third point?
4. What is the formula for the slope of a line?
5. How do you find the area of a triangle formed by three points?

**Tip:** When working with the distance formula, always ensure to subtract the coordinates in the same order (e.g., \(x_2 - x_1\) and \(y_2 - y_1\)) to avoid mistakes.

Learn how to find the distance between two points (1, 1) and (-8, -3) on a coordinate plane using the distance formula. This problem involves applying the distance formula step-by-step, suitable for high school students.

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