To find the distance between two points [tex]\((-3, 4)\)[/tex] and [tex]\((6, -2)\)[/tex], we use the distance formula, which is given by:
[tex]\[ \text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \][/tex]
First, identify the coordinates of the points:
[tex]\((x_1, y_1) = (-3, 4)\)[/tex] and [tex]\((x_2, y_2) = (6, -2)\)[/tex].
Let's calculate the differences in the x-coordinates and the y-coordinates:
[tex]\[ x_2 - x_1 = 6 - (-3) = 6 + 3 = 9 \][/tex]
[tex]\[ y_2 - y_1 = -2 - 4 = -6 \][/tex]
Next, we square these differences:
[tex]\[ (x_2 - x_1)^2 = 9^2 = 81 \][/tex]
[tex]\[ (y_2 - y_1)^2 = (-6)^2 = 36 \][/tex]
Add these squared differences together:
[tex]\[ (x_2 - x_1)^2 + (y_2 - y_1)^2 = 81 + 36 = 117 \][/tex]
Then take the square root of the sum to find the distance:
[tex]\[ \text{Distance} = \sqrt{117} \approx 10.82 \][/tex]
Now, carefully examine the given options:
A. [tex]\( (-3-6)^2 + (4+2)^2 \)[/tex]
It represents the expression for the sum of squared differences but not in the format of the distance formula.
B. [tex]\( (-3-4)^2 + (6+2)^2 \)[/tex]
This calculation uses incorrect coordinates.
C. [tex]\( \sqrt{(-3-6)^2 + (4+2)^2} \)[/tex]
This matches our calculation steps correctly:
[tex]\[ (-3-6)^2 = (-3-6)^2 = 9^2 = 81 \][/tex]
[tex]\[ (4+2)^2 = 6^2 = 36 \][/tex]
[tex]\[ \sqrt{81 + 36} = \sqrt{117} \approx 10.82 \][/tex]
D. [tex]\( \sqrt{(-3-4)^2 + (6+2)^2} \)[/tex]
This option uses incorrect coordinates.
Based on the correct application of the distance formula, the correct option is:
[tex]\[ \boxed{\sqrt{(-3-6)^2 + (4+2)^2}} \][/tex]
So, the correct answer is option C.
