Answer:
The circumcenter of the triangle with vertices A(2,6), B(8,6), and C(8,10) is (8,6).
Explanation:
To find the circumcenter of the triangle with vertices A(2,6), B(8,6), and C(8,10), we can use the following steps:
Step 1: Find the midpoint of two sides
We first find the midpoint of two sides of the triangle. Let's take sides AB and BC:
Midpoint of AB: ((2 + 8)/2, (6 + 6)/2) = (5, 6)
Midpoint of BC: ((8 + 8)/2, (6 + 10)/2) = (8, 8)
Step 2: Find the slope of two sides
Next, we find the slope of the two sides AB and BC:
Slope of AB: (6 - 6)/(8 - 2) = 0
Slope of BC: (10 - 6)/(8 - 8) = undefined
Step 3: Find the perpendicular bisectors of two sides
Since the slope of AB is 0, its perpendicular bisector is a horizontal line passing through the midpoint of AB, which is y=6. Since the slope of BC is undefined, its perpendicular bisector is a vertical line passing through the midpoint of BC, which is x=8.
Step 4: Find the intersection of perpendicular bisectors
The circumcenter is the point where the two perpendicular bisectors intersect. The intersection point is (8,6).
Therefore, the circumcenter of the triangle with vertices A(2,6), B(8,6), and C(8,10) is (8,6).