Let's determine which expression gives the correct distance between the points [tex]\((-3, 4)\)[/tex] and [tex]\((6, -2)\)[/tex].
1. Identify Coordinates and Differences:
- The coordinates of the first point [tex]\((x_1, y_1)\)[/tex] are [tex]\((-3, 4)\)[/tex].
- The coordinates of the second point [tex]\((x_2, y_2)\)[/tex] are [tex]\((6, -2)\)[/tex].
2. Calculate the Differences:
- The difference in the x-coordinates [tex]\(dx\)[/tex] is [tex]\(x_2 - x_1 = 6 - (-3) = 6 + 3 = 9\)[/tex].
- The difference in the y-coordinates [tex]\(dy\)[/tex] is [tex]\(y_2 - y_1 = -2 - 4 = -2 - 4 = -6\)[/tex].
3. Square the Differences:
- Squaring the difference in x-coordinates: [tex]\(dx^2 = 9^2 = 81\)[/tex].
- Squaring the difference in y-coordinates: [tex]\(dy^2 = -6^2 = 36\)[/tex].
4. Sum of the Squared Differences:
- Adding these values: [tex]\(dx^2 + dy^2 = 81 + 36 = 117\)[/tex].
5. Taking the Square Root of the Sum:
- The distance [tex]\(d\)[/tex] is obtained by taking the square root of the sum of the squared differences: [tex]\(\sqrt{117} \approx 10.816653826391969\)[/tex].
6. Determine the Correct Expression:
- Option A: [tex]\((-3-6)^2+(4+2)^2 = (-9)^2 + (6)^2 = 81 + 36\)[/tex], but this is just the sum of the squared differences, not the actual distance (no square root).
- Option B: [tex]\((-3-4)^2+(6+2)^2 = (-7)^2 + (8)^2 = 49 + 64\)[/tex], which calculates to 113 and is incorrect.
- Option C: [tex]\(\sqrt{(-3-6)^2+(4+2)^2} = \sqrt{(-9)^2 + (6)^2} = \sqrt{81 + 36} = \sqrt{117} \approx 10.816653826391969\)[/tex], which matches our calculated distance and is thus correct.
- Option D: [tex]\(\sqrt{(-3-4)^2+(6+2)^2} = \sqrt{(-7)^2 + (8)^2} = \sqrt{49 + 64} = \sqrt{113}\)[/tex], which is incorrect.
Therefore, the correct expression that gives the distance between the points [tex]\((-3,4)\)[/tex] and [tex]\((6,-2)\)[/tex] is:
[tex]\[ \boxed{C} \: \sqrt{(-3-6)^2+(4+2)^2} \][/tex]
