To find the distance between the points [tex]\((5, -3)\)[/tex] and [tex]\((8, 1)\)[/tex], we will use the Euclidean distance formula. The formula for the distance [tex]\(d\)[/tex] between two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] in a 2-dimensional plane is given by:
[tex]\[
d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\][/tex]
Here, we are given the points [tex]\((5, -3)\)[/tex] and [tex]\((8, 1)\)[/tex]. Let’s proceed step by step to find the distance.
1. Identify the coordinates:
- Point 1: [tex]\((x_1, y_1) = (5, -3)\)[/tex]
- Point 2: [tex]\((x_2, y_2) = (8, 1)\)[/tex]
2. Calculate the differences in the x and y coordinates:
- Difference in x-coordinates ([tex]\(\Delta x\)[/tex]):
[tex]\[
\Delta x = x_2 - x_1 = 8 - 5 = 3
\][/tex]
- Difference in y-coordinates ([tex]\(\Delta y\)[/tex]):
[tex]\[
\Delta y = y_2 - y_1 = 1 - (-3) = 1 + 3 = 4
\][/tex]
3. Square the differences:
- [tex]\((\Delta x)^2 = 3^2 = 9\)[/tex]
- [tex]\((\Delta y)^2 = 4^2 = 16\)[/tex]
4. Sum the squares of the differences:
[tex]\[
(\Delta x)^2 + (\Delta y)^2 = 9 + 16 = 25
\][/tex]
5. Take the square root of the sum to find the distance:
[tex]\[
d = \sqrt{25} = 5.0
\][/tex]
Therefore, the distance between the points [tex]\((5, -3)\)[/tex] and [tex]\((8, 1)\)[/tex] is [tex]\(5.0\)[/tex].
