Let's examine the given statement about the distance between the two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((4,8)\)[/tex] to determine if it's true or false.
The distance between two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is calculated using the distance formula:
[tex]\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \][/tex]
In this case, the points we are given are [tex]\((x_1, y_1)\)[/tex] and [tex]\((4, 8)\)[/tex]. Plugging these values into the distance formula, we get:
[tex]\[ d = \sqrt{(4 - x_1)^2 + (8 - y_1)^2} \][/tex]
Now let's compare this with the given expression:
[tex]\[
\sqrt{(x_1 - 8)^2 + (y_1 - 4)^2}
\][/tex]
According to properties of squares, we know that the expressions [tex]\((4 - x_1)^2\)[/tex] and [tex]\((x_1 - 8)^2\)[/tex] are equivalent because squaring a number negates the sign of any difference:
[tex]\[ (4 - x_1)^2 = (x_1 - 4)^2 \][/tex]
Similarly, for the second expression:
[tex]\[ (8 - y_1)^2 = (y_1 - 8)^2 \][/tex]
Therefore:
[tex]\[ \sqrt{(4 - x_1)^2 + (8 - y_1)^2} = \sqrt{(x_1 - 8)^2 + (y_1 - 4)^2} \][/tex]
Since both forms of the distance equations match, the given statement is true. Therefore, the correct answer is:
A. True
