Question

Find the distance between the points (4,0) and (10,2).
Write your answer as a whole number or a fully simplified radical expression. Do not round.

Answer 1

To find the distance between the points \((4,0)\) and \((10,2)\), we can use the distance formula given by:

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

Here, \((x_1, y_1) = (4, 0)\) and \((x_2, y_2) = (10, 2)\). Now we will substitute in the coordinates:

1. Calculate \(x_2 - x_1\):
\[
x_2 - x_1 = 10 - 4 = 6
\]

2. Calculate \(y_2 - y_1\):
\[
y_2 - y_1 = 2 - 0 = 2
\]

3. Substitute these values into the distance formula:
\[
d = \sqrt{(6)^2 + (2)^2}
\]

4. Calculate \(6^2\) and \(2^2\):
\[
6^2 = 36 \quad \text{and} \quad 2^2 = 4
\]

5. Now, add these squares together:
\[
36 + 4 = 40
\]

6. Finally, compute the square root:
\[
d = \sqrt{40}
\]

7. Simplifying \(\sqrt{40}\):
\[
\sqrt{40} = \sqrt{4 \times 10} = \sqrt{4} \cdot \sqrt{10} = 2\sqrt{10}
\]

Therefore, the distance between the points \((4,0)\) and \((10,2)\) is:

\[
\boxed{2\sqrt{10}}
\]