Given the problem, we need to determine whether the distance between the points [tex]\((1,2)\)[/tex] and [tex]\((x_1, y_1)\)[/tex] is correctly represented by the formula [tex]\(\sqrt{(x_1 - 1)^2 + (y_1 - 2)^2}\)[/tex].
We will use the distance formula for two points in a Cartesian plane. The distance [tex]\(d\)[/tex] between two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is given by:
[tex]\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \][/tex]
In our problem:
- The coordinates of the first point [tex]\((x_2, y_2)\)[/tex] are [tex]\((1, 2)\)[/tex].
- The coordinates of the second point [tex]\((x_1, y_1)\)[/tex] are [tex]\((x_1, y_1)\)[/tex].
Substituting these coordinates into the distance formula, we get:
[tex]\[ d = \sqrt{(x_1 - 1)^2 + (y_1 - 2)^2} \][/tex]
Thus, the given distance formula is:
[tex]\[ \sqrt{(x_1 - 1)^2 + (y_1 - 2)^2} \][/tex]
Now, we need to check if this formula is correct for the distance between the points [tex]\((1,2)\)[/tex] and [tex]\((x_1, y_1)\)[/tex]. The formula we derived and the given formula are indeed the same:
[tex]\[ \sqrt{(x_1 - 1)^2 + (y_1 - 2)^2} \][/tex]
Therefore, the given statement about the distance formula is:
A. True
