To determine the length of the radius of the circle represented by the equation [tex]\((x + 6)^2 + (y + 8)^2 = 9\)[/tex], we need to compare it to the standard form of a circle's equation. The standard form 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.
Here's the equation again for reference:
[tex]\[
(x + 6)^2 + (y + 8)^2 = 9
\][/tex]
In this equation:
- [tex]\(h = -6\)[/tex] (since [tex]\(x + 6\)[/tex] can be rewritten as [tex]\(x - (-6)\)[/tex])
- [tex]\(k = -8\)[/tex] (since [tex]\(y + 8\)[/tex] can be rewritten as [tex]\(y - (-8)\)[/tex])
- [tex]\(r^2 = 9\)[/tex]
To find the radius [tex]\(r\)[/tex], we need to take the square root of [tex]\(r^2\)[/tex]:
[tex]\[
r = \sqrt{9}
\][/tex]
Calculating the square root of 9:
[tex]\[
r = 3
\][/tex]
Therefore, the length of the radius of circle [tex]\(F\)[/tex] is:
\[
\boxed{3}
