Find the distance between the two points in simplest radical form.
left parenthesis, 9, comma, minus, 1, right parenthesis, and , left parenthesis, 3, comma, 3, right parenthesis
(9,−1) and (3,3)

To find the distance between the two points \((9, -1)\) and \((3, 3)\), we use the distance formula:

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

Here, \((x_1, y_1) = (9, -1)\) and \((x_2, y_2) = (3, 3)\). Plugging these coordinates into the formula:

\[
d = \sqrt{(3 - 9)^2 + (3 - (-1))^2}
\]

Simplify the expressions inside the square root:

\[
d = \sqrt{(-6)^2 + (4)^2}
\]

\[
d = \sqrt{36 + 16}
\]

\[
d = \sqrt{52}
\]

Next, simplify \(\sqrt{52}\) by factoring:

\[
d = \sqrt{4 \times 13} = \sqrt{4} \times \sqrt{13} = 2\sqrt{13}
\]

So, the distance between the two points in simplest radical form is \( \boxed{2\sqrt{13}} \).

Would you like more details, or do you have any questions? Here are some related questions for further practice:

1. Find the distance between \((5, -2)\) and \((1, 4)\).
2. Calculate the distance between \((7, 8)\) and \((2, -1)\).
3. What is the distance between \((-3, 6)\) and \((4, -2)\)?
4. Determine the distance between \((0, 0)\) and \((9, 12)\).
5. If two points are \((x_1, y_1)\) and \((x_2, y_2)\), write the general formula to find the distance between them.

**Tip:** Always check if the radical expression can be simplified further by factoring out perfect squares.

Learn how to find the distance between two points (9, -1) and (3, 3) using the distance formula. Step-by-step solution with simplification to simplest radical form. Suitable for students in Grades 7-9.

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