Find the circumcenter of triangle ABC. A is at (2,5), B is at (2,3) and C is located at (6,3).
1. (4,4)
2.(4,2)
3.(5,3)
4.(3,5)
To find the circumcenter of triangle ABC, we can use the properties of perpendicular bisectors.
First, let's find the midpoint of AB. The midpoint of AB is ((2+2)/2, (5+3)/2) = (2,4).
Next, let's find the slope of AB. The slope of AB is (3-5)/(2-2) = -2/0. However, the slope is undefined, which means that AB is a vertical line.
Since AB is vertical, the perpendicular bisector of AB will be a horizontal line passing through the midpoint of AB, which is (2,4).
Similarly, let's find the midpoint of BC. The midpoint of BC is ((2+6)/2, (3+3)/2) = (4,3).
Next, let's find the slope of BC. The slope of BC is (3-3)/(6-2) = 0.
Since BC is horizontal, the perpendicular bisector of BC will be a vertical line passing through the midpoint of BC, which is (4,3).
Now, we can find the intersection point of the perpendicular bisectors, which will be the circumcenter of triangle ABC.
Since the perpendicular bisector of AB is a horizontal line passing through (2,4), and the perpendicular bisector of BC is a vertical line passing through (4,3), the intersection point will be the coordinates (4,3).
Therefore, the circumcenter of triangle ABC is (4,3).
The correct answer is option 3. (5,3)