To find the distance between the points [tex]\((4,6)\)[/tex] and [tex]\((7,-3)\)[/tex], we can use the distance formula. The distance formula for two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is given by:
[tex]\[
\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\][/tex]
Let's determine the correct expression step by step:
1. Subtract the x-coordinates:
[tex]\[
x_2 - x_1 = 7 - 4 = 3
\][/tex]
2. Subtract the y-coordinates:
[tex]\[
y_2 - y_1 = -3 - 6 = -9
\][/tex]
3. Square the differences:
[tex]\[
(x_2 - x_1)^2 = 3^2 = 9
\][/tex]
[tex]\[
(y_2 - y_1)^2 = (-9)^2 = 81
\][/tex]
4. Add the squared differences:
[tex]\[
9 + 81 = 90
\][/tex]
5. Take the square root of the sum:
[tex]\[
\sqrt{90}
\][/tex]
The expression that matches this step-by-step solution is:
[tex]\[
\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} = \sqrt{(4 - 7)^2 + (6 - 3)^2}
\][/tex]
Among the given options, the correct one that represents the distance formula is:
D. [tex]\(\sqrt{(4-7)^2+(6-3)^2}\)[/tex]
