To find the distance between the points [tex]\((1, -2)\)[/tex] and [tex]\((2, 4)\)[/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 [tex]\((x_1, y_1) = (1, -2)\)[/tex] and [tex]\((x_2, y_2) = (2, 4)\)[/tex].
Now, compute the differences in the [tex]\(x\)[/tex]-coordinates and [tex]\(y\)[/tex]-coordinates:
[tex]\[
(x_2 - x_1) = 2 - 1 = 1
\][/tex]
[tex]\[
(y_2 - y_1) = 4 - (-2) = 6
\][/tex]
Next, square these differences:
[tex]\[
(x_2 - x_1)^2 = 1^2 = 1
\][/tex]
[tex]\[
(y_2 - y_1)^2 = 6^2 = 36
\][/tex]
Now, sum the squared differences:
[tex]\[
(x_2 - x_1)^2 + (y_2 - y_1)^2 = 1 + 36 = 37
\][/tex]
Finally, take the square root of this sum to find the distance:
[tex]\[
\text{Distance} = \sqrt{37} \approx 6.082762530298219
\][/tex]
Given the expressions in the options:
- Option A. [tex]\((1+2)^2 + (2-4)^2\)[/tex] is incorrect because it incorrectly sums [tex]\(x\)[/tex]-coordinates and does not use the correct difference.
- Option C. [tex]\((1 - 2)^2 + (-2 - 4)^2\)[/tex] misses the square root and thus does not give the distance.
- Option D. [tex]\(\sqrt{(1+2)^2 + (2 - 4)^2}\)[/tex] incorrectly sums [tex]\(x\)[/tex] coordinates.
The correct option is B, which matches the distance formula:
[tex]\[
\sqrt{(1 - 2)^2 + (-2 - 4)^2}
\][/tex]
Thus, the correct answer is:
[tex]\[
\boxed{\sqrt{(1 - 2)^2 + (-2 - 4)^2}}
\][/tex]
