Sure! To determine the radius of a circle centered at [tex]\((5, -7)\)[/tex] that passes through the point [tex]\((10, 5)\)[/tex], you can use the distance formula. The distance formula is applied to find the distance between two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] in a Cartesian plane, and it is given by:
[tex]\[
d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\][/tex]
Here:
- The center of the circle [tex]\((x_1, y_1)\)[/tex] is [tex]\((5, -7)\)[/tex].
- The point on the circle [tex]\((x_2, y_2)\)[/tex] is [tex]\((10, 5)\)[/tex].
Plugging these coordinates into the distance formula, we get:
[tex]\[
d = \sqrt{(10 - 5)^2 + (5 + 7)^2}
\][/tex]
Simplifying inside the parentheses first:
[tex]\[
d = \sqrt{(5)^2 + (12)^2}
\][/tex]
Then, squaring the numbers inside the square root:
[tex]\[
d = \sqrt{25 + 144}
\][/tex]
Next, adding the results from within the square root:
[tex]\[
d = \sqrt{169}
\][/tex]
Finally, taking the square root of 169:
[tex]\[
d = 13
\][/tex]
So, the radius of the circle is [tex]\(\boxed{13}\)[/tex].
