To find the distance between the two points [tex]\((1, 3)\)[/tex] and [tex]\((4, -1)\)[/tex], we can use the distance formula. The distance formula calculates the straight-line distance between two points in a plane and is given by:
[tex]\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \][/tex]
Let's go through the steps to find this distance:
1. Identify the coordinates of the points:
- Point 1: [tex]\((x_1, y_1) = (1, 3)\)[/tex]
- Point 2: [tex]\((x_2, y_2) = (4, -1)\)[/tex]
2. Calculate the differences in [tex]\(x\)[/tex] and [tex]\(y\)[/tex] coordinates:
- Difference in [tex]\(x\)[/tex] coordinates:
[tex]\[
\Delta x = x_2 - x_1 = 4 - 1 = 3
\][/tex]
- Difference in [tex]\(y\)[/tex] coordinates:
[tex]\[
\Delta y = y_2 - y_1 = -1 - 3 = -4
\][/tex]
3. Square the differences:
- Square of the difference in [tex]\(x\)[/tex] coordinates:
[tex]\[
(\Delta x)^2 = 3^2 = 9
\][/tex]
- Square of the difference in [tex]\(y\)[/tex] coordinates:
[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]\[
d = \sqrt{25} = 5
\][/tex]
Therefore, the distance between the points [tex]\((1, 3)\)[/tex] and [tex]\((4, -1)\)[/tex] is [tex]\(5\)[/tex].
The correct answer is [tex]\(\boxed{5}\)[/tex].
