To find the distance between the points [tex]\((1, -2)\)[/tex] and [tex]\((2, 4)\)[/tex], we can use the Euclidean distance formula. The Euclidean distance formula between two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is:
[tex]\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \][/tex]
First, let's identify the coordinates of the points:
- [tex]\((x_1, y_1) = (1, -2)\)[/tex]
- [tex]\((x_2, y_2) = (2, 4)\)[/tex]
Next, we plug these values into the distance formula:
[tex]\[ d = \sqrt{(2 - 1)^2 + (4 - (-2))^2} \][/tex]
To simplify this, let's calculate each term:
1. Calculate the difference in [tex]\(x\)[/tex]-coordinates and square it:
[tex]\[ (2 - 1)^2 = 1^2 = 1 \][/tex]
2. Calculate the difference in [tex]\(y\)[/tex]-coordinates and square it:
[tex]\[ (4 - (-2))^2 = (4 + 2)^2 = 6^2 = 36 \][/tex]
Next, sum the squared differences:
[tex]\[ 1 + 36 = 37 \][/tex]
Finally, take the square root of the sum to find the distance:
[tex]\[ d = \sqrt{37} \approx 6.082762530298219 \][/tex]
Thus, the expression that gives the distance between the points [tex]\((1, -2)\)[/tex] and [tex]\((2, 4)\)[/tex] is:
[tex]\[ \sqrt{(1 - 2)^2 + (-2 - 4)^2} \][/tex]
Therefore, the correct answer is:
[tex]\[ \boxed{\text{D. } \sqrt{(1-2)^2+(-2-4)^2}} \][/tex]
