Answer:
[tex](\frac{1}{2}, \frac{-1}{2})[/tex] is the right option.
Step-by-step explanation:
Let us consider the triangle ΔVUW.
The given figure shows that the angle at vertex V is 90°. So, it is a right triangle. So, the side UW will be the hypotenuse of the right triangle ΔVUW.
If we draw perpendicular bisectors of the three sides of this right triangle, the hypotenuse UW will have a midpoint at which these perpendicular bisectors would meet. This midpoint on hypotenuse UW would be the circumcenter of a this right angle triangle ΔVUW.
If we circumscribe a circle around this right triangle, this triangle will have circumcenter, which would also become the center of that circle.
This hypotenuse UW could be also treated as diameter of the circle, as the center of circumscribed center would lie on hypotenuse UW.
The coordinates of U is (-3 ,2) and the coordinates of W is (4,-3).
Midpoint of UW could be determined by this formula:
[tex](\frac{x_{1} + x_{2}}{2},\frac{y_{1} + y_{2}}{2})[/tex]
So,
Midpoint of UW = [tex](\frac{-3 + 4}{2},\frac{2 + (-3)}{2})[/tex]
Midpoint of UW = [tex](\frac{1}{2},\frac{-1}{2})[/tex]
So, the coordinates of circumcenter will be: [tex](\frac{1}{2}, \frac{-1}{2})[/tex]
As we know that If we circumscribe a circle around this right triangle, this triangle will have circumcenter, which would also become the center of that circle.
Hence,
The coordinates of circumcenter will also be the coordinate of circumference of the circle which would be [tex](\frac{1}{2}, \frac{-1}{2})[/tex].
So, [tex](\frac{1}{2}, \frac{-1}{2})[/tex] is the right option.
Keywords: circumcenter, triangle, circumference, perpendicular bisector
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