To find the distance between the two points [tex]\((8, -3)\)[/tex] and [tex]\((4, -7)\)[/tex], we can utilize the distance formula. The distance formula between two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] in the coordinate plane is given by:
[tex]\[ \text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \][/tex]
Let's apply this formula step-by-step to the points [tex]\((8, -3)\)[/tex] and [tex]\((4, -7)\)[/tex].
1. Subtract the x-coordinates:
[tex]\[ x_2 - x_1 = 4 - 8 = -4 \][/tex]
2. Subtract the y-coordinates:
[tex]\[ y_2 - y_1 = -7 - (-3) = -7 + 3 = -4 \][/tex]
3. Square the differences:
[tex]\[ (-4)^2 = 16 \][/tex]
[tex]\[ (-4)^2 = 16 \][/tex]
4. Add the squared differences:
[tex]\[ 16 + 16 = 32 \][/tex]
5. Take the square root of the sum:
[tex]\[ \sqrt{32} \][/tex]
Thus, the distance between points [tex]\((8, -3)\)[/tex] and [tex]\((4, -7)\)[/tex] is [tex]\(\sqrt{32}\)[/tex].
From the given options, the correct answer is:
(C) [tex]\(\sqrt{32}\)[/tex]
