To determine which expression gives the distance between the points [tex]\((1, -2)\)[/tex] and [tex]\((2, 4)\)[/tex], we utilize the distance formula between two points in a plane. The distance formula is given by
[tex]\[
\text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\][/tex]
Given points are [tex]\( (1, -2) \)[/tex] for [tex]\((x_1, y_1)\)[/tex] and [tex]\( (2, 4) \)[/tex] for [tex]\((x_2, y_2)\)[/tex]. Plugging in these values, we get:
[tex]\[
\text{Distance} = \sqrt{(2 - 1)^2 + (4 - (-2))^2}
\][/tex]
Now, let's simplify inside the square root:
1. Calculate [tex]\( (2 - 1)^2 \)[/tex]:
[tex]\[
(2 - 1)^2 = 1^2 = 1
\][/tex]
2. Calculate [tex]\( (4 - (-2))^2 \)[/tex]:
[tex]\[
4 - (-2) = 4 + 2 = 6
\][/tex]
[tex]\[
6^2 = 36
\][/tex]
Adding these values together inside the square root:
[tex]\[
1 + 36 = 37
\][/tex]
So, the distance becomes:
[tex]\[
\text{Distance} = \sqrt{37} \approx 6.082762530298219
\][/tex]
According to this detailed derivation, the correct expression that calculates this distance is:
[tex]\[
\sqrt{(1-2)^2 + (-2-4)^2}
\][/tex]
Therefore, the correct answer is:
C. [tex]\(\sqrt{(1-2)^2 + (-2-4)^2}\)[/tex]
