To determine which expression gives the distance between the points [tex]\((-3, 4)\)[/tex] and [tex]\((6, -2)\)[/tex], we can use the Euclidean distance formula. The distance [tex]\( d \)[/tex] between two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] in a Cartesian plane is given by:
[tex]\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \][/tex]
### Step-by-step solution:
1. Identify the coordinates:
- Point 1: [tex]\((-3, 4)\)[/tex]
- Point 2: [tex]\((6, -2)\)[/tex]
2. Calculate the differences in the coordinates:
- Difference in [tex]\( x \)[/tex]: [tex]\( x_2 - x_1 = 6 - (-3) = 6 + 3 = 9 \)[/tex]
- Difference in [tex]\( y \)[/tex]: [tex]\( y_2 - y_1 = -2 - 4 = -6 \)[/tex]
3. Substitute these differences into the Euclidean distance formula:
[tex]\[
d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} = \sqrt{(9)^2 + (-6)^2} = \sqrt{81 + 36} = \sqrt{117}
\][/tex]
4. Now match this expression with the given options:
- Option A: [tex]\((-3 - 6)^2 + (4 + 2)^2\)[/tex]
[tex]\[
= (-9)^2 + (6)^2
= 81 + 36 = 117 \quad \text{(sum of squared distances, but without square root)}
\][/tex]
- Option B: [tex]\(\sqrt{(-3 - 4)^2 + (6 + 2)^2}\)[/tex]
[tex]\[
= \sqrt{(-7)^2 + (8)^2}
= \sqrt{49 + 64}
= \sqrt{113} \quad \text{(incorrect differences calculated)}
\][/tex]
- Option C: [tex]\(\sqrt{(-3 - 6)^2 + (4 + 2)^2}\)[/tex]
[tex]\[
= \sqrt{(-9)^2 + (6)^2}
= \sqrt{81 + 36}
= \sqrt{117} \quad \text{(correct expression for distance)}
\][/tex]
- Option D: [tex]\((-3 - 4)^2 + (6 + 2)^2\)[/tex]
[tex]\[
= (-7)^2 + (8)^2
= 49 + 64 = 113 \quad \text{(sum of squared distances, but without square root)}
\][/tex]
Therefore, the expression that correctly gives the distance between the points [tex]\((-3, 4)\)[/tex] and [tex]\((6, -2)\)[/tex] is:
[tex]\[ \boxed{C} \][/tex]
