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------------------------------------------------ Select the correct answer.

Circle F is represented by the equation [tex](x+6)^2+(y+8)^2=9[/tex]. What is the length of the radius of circle F?

A. 3
B. 9
C. 10
D. 81

Answer :

To determine the length of the radius of the circle represented by the equation [tex]\((x+6)^2+(y+8)^2=9\)[/tex], we can use the standard form of the equation of a circle. The general form of a circle's equation is:

[tex]\[
(x - h)^2 + (y - k)^2 = r^2
\][/tex]

Here, [tex]\((h, k)\)[/tex] represents the center of the circle, and [tex]\(r\)[/tex] represents the radius.

1. Identify the center and radius:
- From the given equation [tex]\((x + 6)^2 + (y + 8)^2 = 9\)[/tex], we can observe that:
- [tex]\((x + 6)^2\)[/tex] can be rewritten as [tex]\((x - (-6))^2\)[/tex]. Thus, [tex]\(h = -6\)[/tex].
- [tex]\((y + 8)^2\)[/tex] can be rewritten as [tex]\((y - (-8))^2\)[/tex]. Thus, [tex]\(k = -8\)[/tex].

Therefore, the center of the circle is [tex]\((-6, -8)\)[/tex].

2. Determine the radius:
- The right side of the equation, [tex]\(9\)[/tex], represents [tex]\(r^2\)[/tex] (the square of the radius).
- To find the radius [tex]\(r\)[/tex], we need to take the square root of [tex]\(9\)[/tex].

3. Calculate the radius:
[tex]\[
r = \sqrt{9} = 3
\][/tex]

Thus, the length of the radius of circle [tex]\(F\)[/tex] is [tex]\(3\)[/tex].

So, the correct answer is:
[tex]\[
\boxed{3}
\][/tex]