To determine the distance between two points in a Cartesian plane, we can use the distance formula:
[tex]\[ \text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \][/tex]
We are given two points: [tex]\((1, -2)\)[/tex] and [tex]\((2, 4)\)[/tex].
1. Identify the coordinates:
[tex]\[
(x_1, y_1) = (1, -2) \quad \text{and} \quad (x_2, y_2) = (2, 4)
\][/tex]
2. Calculate the difference in the [tex]\(x\)[/tex]-coordinates:
[tex]\[
x_2 - x_1 = 2 - 1 = 1
\][/tex]
Then, square the difference:
[tex]\[
(x_2 - x_1)^2 = 1^2 = 1
\][/tex]
3. Calculate the difference in the [tex]\(y\)[/tex]-coordinates:
[tex]\[
y_2 - y_1 = 4 - (-2) = 4 + 2 = 6
\][/tex]
Then, square the difference:
[tex]\[
(y_2 - y_1)^2 = 6^2 = 36
\][/tex]
4. Add the squared differences:
[tex]\[
(x_2 - x_1)^2 + (y_2 - y_1)^2 = 1 + 36
\][/tex]
5. Take the square root of the sum to find the distance:
[tex]\[
\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} = \sqrt{1 + 36} = \sqrt{37}
\][/tex]
Therefore, the expression that correctly represents the distance between the points [tex]\((1, -2)\)[/tex] and [tex]\((2, 4)\)[/tex] is:
C. [tex]\(\sqrt{(1-2)^2 + (-2-4)^2}\)[/tex]
The simplified answer to this expression gives us a distance of [tex]\(\sqrt{37}\)[/tex], and this corresponds to choice C, which is the correct expression for calculating the distance between the points [tex]\((1, -2)\)[/tex] and [tex]\((2, 4)\)[/tex].
