Final answer:
The circumcenter of triangle EFG with vertices E(3, 6), F(3, 4), and G(7, 4) is the point where the perpendicular bisectors of the sides intersect, which is at (5, 5). The correct answer is a) (5, 5).
Step-by-step explanation:
The question is on finding the cicumcenter of a triangle. The circumcenter of a triangle is the point where the perpendicular bisectors of the sides of the triangle intersect.
This single point is equidistant from all the vertices of the triangle. For triangle EFG, with vertices E(3, 6), F(3, 4), and G(7, 4), we can observe that FG is a horizontal line segment and EF is a vertical line segment.
Since FG is horizontal, its perpendicular bisector will be a vertical line passing through the midpoint of FG, which is (5, 4). Likewise, since EF is vertical, its perpendicular bisector will be a horizontal line passing through the midpoint of EF, which is (3, 5).
The point where these two perpendicular bisectors would intersect would be at the horizontal level of the midpoint of EF and the vertical level of the midpoint of FG, thus giving us the circumcenter at (5, 5), which corresponds to option (a).