Find the distance between the points (6,5) and (10,1).
Write your answer as a whole number or a fully simplified radical expression. Do not round.

To find the distance between the two points \((6,5)\) and \((10,1)\), we can use the distance formula. The distance formula for two points \((x_1, y_1)\) and \((x_2, y_2)\) is:

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

Substituting the given points \((6,5)\) and \((10,1)\) into the formula, we identify:

- \(x_1 = 6\), \(y_1 = 5\)
- \(x_2 = 10\), \(y_2 = 1\)

Now plug these values into the formula:

\[
d = \sqrt{(10 - 6)^2 + (1 - 5)^2}
\]

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

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

\[
d = \sqrt{32}
\]

\[
d = \sqrt{16 \times 2}
\]

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

So, the distance between the points \((6,5)\) and \((10,1)\) is \(4\sqrt{2}\).

Would you like more details or have any questions?

### Related Questions:
1. How would the distance formula change if the points were in three-dimensional space?
2. What is the distance between the points \((2,3)\) and \((7,9)\)?
3. If the distance between two points is \(4\sqrt{2}\), can we determine their coordinates?
4. How can you verify that \(4\sqrt{2}\) is the simplest radical form of \(\sqrt{32}\)?
5. What would the distance be between two points that lie on the same line?

### Tip:
Always double-check your calculations when working with radicals to ensure the expression is fully simplified.

Learn how to calculate the distance between two points (6,5) and (10,1) using the distance formula. This problem involves applying the distance formula in coordinate geometry to find the distance between given points. Suitable for students in Grades 8-10.

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