Which expression gives the distance between the points [tex]$(1, -2)$[/tex] and [tex]$(2, 4)$[/tex]?

A. [tex]$(1 + 2)^2 + (2 - 4)^2$[/tex]
B. [tex]$\sqrt{(1 - 2)^2 + (-2 - 4)^2}$[/tex]
C. [tex][tex]$(1 - 2)^2 + (-2 - 4)^2$[/tex][/tex]
D. [tex]$\sqrt{(1 + 2)^2 + (2 - 4)^2}$[/tex]



Answer :

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]