Let's find the distance between the two points [tex]\((0,3)\)[/tex] and [tex]\((4,0)\)[/tex] step by step.
1. Identify the coordinates of the points:
- Point 1: [tex]\((x_1, y_1) = (0, 3)\)[/tex]
- Point 2: [tex]\((x_2, y_2) = (4, 0)\)[/tex]
2. Calculate the changes in the [tex]\(x\)[/tex]-coordinates and [tex]\(y\)[/tex]-coordinates:
- Change in [tex]\(x\)[/tex]-coordinates: [tex]\(\Delta x = x_2 - x_1 = 4 - 0 = 4\)[/tex]
- Change in [tex]\(y\)[/tex]-coordinates: [tex]\(\Delta y = y_2 - y_1 = 0 - 3 = -3\)[/tex]
3. Calculate the distance using the Pythagorean theorem:
- The distance [tex]\(d\)[/tex] between two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] in the Cartesian plane is given by:
[tex]\[
d = \sqrt{(\Delta x)^2 + (\Delta y)^2}
\][/tex]
4. Substitute the values of [tex]\(\Delta x\)[/tex] and [tex]\(\Delta y\)[/tex]:
- [tex]\((\Delta x)^2 = 4^2 = 16\)[/tex]
- [tex]\((\Delta y)^2 = (-3)^2 = 9\)[/tex]
5. Compute the sum of squares:
- Sum of squares: [tex]\(16 + 9 = 25\)[/tex]
6. Find the square root of the sum of squares to get the distance:
- Distance [tex]\(d = \sqrt{25} = 5.0\)[/tex]
Hence, the distance between the points [tex]\((0, 3)\)[/tex] and [tex]\((4, 0)\)[/tex] is [tex]\(5.0\)[/tex].
