To find the distance between the points [tex]\((4, 6)\)[/tex] and [tex]\((7, -3)\)[/tex], we use the distance formula, which is defined as:
[tex]\[ \text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \][/tex]
Here, [tex]\((x_1, y_1) = (4, 6)\)[/tex] and [tex]\((x_2, y_2) = (7, -3)\)[/tex].
Let's proceed step-by-step:
1. Calculate the difference in the [tex]\(x\)[/tex]-coordinates:
[tex]\[ x_2 - x_1 = 7 - 4 = 3 \][/tex]
Then square this difference:
[tex]\[ (x_2 - x_1)^2 = 3^2 = 9 \][/tex]
2. Calculate the difference in the [tex]\(y\)[/tex]-coordinates:
[tex]\[ y_2 - y_1 = -3 - 6 = -9 \][/tex]
Then square this difference:
[tex]\[ (y_2 - y_1)^2 = (-9)^2 = 81 \][/tex]
3. Sum these squared differences:
[tex]\[ (x_2 - x_1)^2 + (y_2 - y_1)^2 = 9 + 81 = 90 \][/tex]
4. Take the square root of the sum to get the distance:
[tex]\[ \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} = \sqrt{90} \approx 9.4868 \][/tex]
Given the provided multiple-choice options, the correct expression that matches our calculations and gives the distance is:
[tex]\[ \text{Option B: } \sqrt{(4-7)^2 + (6+3)^2} \][/tex]
Thus, the correct answer is:
[tex]\[ \boxed{\sqrt{(4-7)^2 + (6+3)^2}} \][/tex]
