To determine the distance between the points [tex]\((4, 3)\)[/tex] and [tex]\((1, -1)\)[/tex] on the coordinate plane, we can use the distance formula. The distance formula is derived from the Pythagorean theorem and is given as:
[tex]\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \][/tex]
Here, [tex]\((x_1, y_1) = (4, 3)\)[/tex] and [tex]\((x_2, y_2) = (1, -1)\)[/tex].
Let's follow the steps to calculate the distance:
1. Calculate the difference in the x-coordinates:
[tex]\[ x_2 - x_1 = 1 - 4 = -3 \][/tex]
2. Calculate the difference in the y-coordinates:
[tex]\[ y_2 - y_1 = -1 - 3 = -4 \][/tex]
3. Square the differences:
[tex]\[ (-3)^2 = 9 \][/tex]
[tex]\[ (-4)^2 = 16 \][/tex]
4. Add the squares of the differences:
[tex]\[ 9 + 16 = 25 \][/tex]
5. Take the square root of the sum:
[tex]\[ \sqrt{25} = 5 \][/tex]
Thus, the distance between the points [tex]\((4, 3)\)[/tex] and [tex]\((1, -1)\)[/tex] is [tex]\(5\)[/tex] units. Therefore, the correct answer is:
D. 5 units
