To determine if the statement is true or false, let’s use the Euclidean distance formula between two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] in a plane. The Euclidean distance [tex]\(d\)[/tex] is given by:
[tex]\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \][/tex]
In this problem, we have the points [tex]\((x_1, y_1)\)[/tex] and [tex]\((3, 7)\)[/tex]. So, [tex]\(x_2 = 3\)[/tex] and [tex]\(y_2 = 7\)[/tex]. Plugging these into the distance formula, we get:
[tex]\[ d = \sqrt{(3 - x_1)^2 + (7 - y_1)^2} \][/tex]
To simplify, consider:
[tex]\[ (3 - x_1)^2 = (x_1 - 3)^2 \][/tex]
[tex]\[ (7 - y_1)^2 = (y_1 - 7)^2 \][/tex]
So, we can rewrite the distance as:
[tex]\[ d = \sqrt{(x_1 - 3)^2 + (y_1 - 7)^2} \][/tex]
The statement given is that the distance between the points [tex]\((3, 7)\)[/tex] and [tex]\((x_1, y_1)\)[/tex] is:
[tex]\[ \sqrt{(x_1-3)^2+(y_1-7)^2} \][/tex]
This matches the Euclidean distance formula derived above. Hence, the statement is:
A. True
