To determine the radius [tex]\(r\)[/tex] of the circle described by the equation [tex]\((x-5)^2 + (y+3)^2 = r^2\)[/tex], we need to find the distance between the center of the circle [tex]\(T\)[/tex] and the point [tex]\(S\)[/tex] that lies on the circle.
First, identify the coordinates:
- The center of the circle [tex]\(T\)[/tex] is at [tex]\((5, -3)\)[/tex].
- The point [tex]\(S\)[/tex] is at [tex]\((-1, 6)\)[/tex].
To find the distance between these two points, use the distance formula:
[tex]\[
\text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\][/tex]
Substitute [tex]\( (x_1, y_1) = (5, -3) \)[/tex] and [tex]\( (x_2, y_2) = (-1, 6) \)[/tex]:
[tex]\[
\text{Distance} = \sqrt{((-1) - 5)^2 + (6 - (-3))^2}
\][/tex]
Simplify inside the parentheses:
[tex]\[
= \sqrt{(-6)^2 + (9)^2}
\][/tex]
Square the values:
[tex]\[
= \sqrt{36 + 81}
\][/tex]
Add the squared values:
[tex]\[
= \sqrt{117}
\][/tex]
Simplify the square root:
[tex]\[
= \sqrt{9 \times 13} = 3 \sqrt{13}
\][/tex]
Thus, the radius [tex]\(r\)[/tex] of the circle is [tex]\(3 \sqrt{13}\)[/tex].
So, the length of the radius of circle [tex]\(T\)[/tex] is:
[tex]\[
\boxed{3 \sqrt{13}}
\][/tex]
