To determine the length of the radius of circle [tex]\( O \)[/tex] represented by the equation [tex]\((x+7)^2+(y+7)^2=16\)[/tex], we need to analyze and compare the given equation with the standard form of a circle's equation.
The standard form of a circle's equation is:
[tex]\[
(x - h)^2 + (y - k)^2 = r^2
\][/tex]
where [tex]\((h, k)\)[/tex] is the center of the circle and [tex]\(r\)[/tex] is the radius.
Given the equation:
[tex]\[
(x+7)^2 + (y+7)^2 = 16
\][/tex]
we can rewrite [tex]\(x + 7\)[/tex] as [tex]\(x - (-7)\)[/tex] and [tex]\(y + 7\)[/tex] as [tex]\(y - (-7)\)[/tex]. Thus, it takes the form:
[tex]\[
(x - (-7))^2 + (y - (-7))^2 = 16
\][/tex]
From this equation, it is clear that the center of the circle [tex]\((h, k)\)[/tex] is [tex]\((-7, -7)\)[/tex].
The right side of the equation [tex]\(16\)[/tex] represents [tex]\(r^2\)[/tex]. Therefore,
[tex]\[
r^2 = 16
\][/tex]
To find the radius [tex]\(r\)[/tex], we take the square root of both sides:
[tex]\[
r = \sqrt{16} = 4
\][/tex]
Thus, the length of the radius of circle [tex]\(O\)[/tex] is [tex]\(\boxed{4}\)[/tex].
The correct answer is:
B. 4
