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.