To find the distance between the points [tex]\((2, 5)\)[/tex] and [tex]\((6, 8)\)[/tex], follow these steps:
1. Identify the coordinates of the points:
The first point is [tex]\((x_1, y_1) = (2, 5)\)[/tex].
The second point is [tex]\((x_2, y_2) = (6, 8)\)[/tex].
2. Calculate the differences in the x-coordinates and y-coordinates:
[tex]\[
\Delta x = x_2 - x_1 = 6 - 2 = 4
\][/tex]
[tex]\[
\Delta y = y_2 - y_1 = 8 - 5 = 3
\][/tex]
3. Apply the distance formula:
The distance [tex]\(d\)[/tex] between two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is given by:
[tex]\[
d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\][/tex]
Substituting the differences calculated:
[tex]\[
d = \sqrt{(4)^2 + (3)^2}
\][/tex]
4. Simplify the expression inside the square root:
[tex]\[
d = \sqrt{16 + 9} = \sqrt{25}
\][/tex]
5. Find the square root of 25:
[tex]\[
d = 5
\][/tex]
So, the distance between the points [tex]\((2, 5)\)[/tex] and [tex]\((6, 8)\)[/tex] is [tex]\(5\)[/tex]. The correct answer from the given choices is [tex]\(5\)[/tex].
