To determine the distance between the two points [tex]\((1,1)\)[/tex] and [tex]\((4,5)\)[/tex] on a grid, we can follow these steps:
1. Identify the coordinates:
The coordinates of the points are [tex]\( (x_1, y_1) = (1, 1) \)[/tex] and [tex]\( (x_2, y_2) = (4, 5) \)[/tex].
2. Calculate the differences in coordinates:
[tex]\[
\Delta x = x_2 - x_1 = 4 - 1 = 3
\][/tex]
[tex]\[
\Delta y = y_2 - y_1 = 5 - 1 = 4
\][/tex]
3. Use the Euclidean distance formula:
The distance [tex]\( d \)[/tex] between two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] on a grid is calculated by:
[tex]\[
d = \sqrt{(\Delta x)^2 + (\Delta y)^2}
\][/tex]
Substituting the values:
[tex]\[
d = \sqrt{(3)^2 + (4)^2}
\][/tex]
[tex]\[
d = \sqrt{9 + 16}
\][/tex]
[tex]\[
d = \sqrt{25}
\][/tex]
[tex]\[
d = 5
\][/tex]
So, the distance between the two points [tex]\((1, 1)\)[/tex] and [tex]\((4, 5)\)[/tex] is [tex]\(5.0\)[/tex] units.
