Answer:
The circumcenter of the triangle with vertices (-7, -1), (-1, -1), and (-7, -9) is (-4, -8).
Explanation:
To find the coordinates of the circumcenter of the triangle with vertices (-7, -1), (-1, -1), and (-7, -9), 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: ((-7 + (-1))/2, (-1 + (-1))/2) = (-4, -1)
Midpoint of BC: ((-1 + (-7))/2, (-1 + (-9))/2) = (-4, -5)
Step 2: Find the slope of two sides
Next, we find the slope of the two sides AB and BC:
Slope of AB: (-1 - (-1))/(-1 - (-7)) = 0/6 = 0
Slope of BC: (-9 - (-1))/(-7 - (-1)) = -8/(-6) = 4/3
Step 3: Find the perpendicular bisectors of two sides
We can now find the equations of the perpendicular bisectors of the two sides AB and BC. Since the slope of the perpendicular bisector is the negative reciprocal of the slope of the side, we have:
Equation of perpendicular bisector of AB:
y - (-1) = (1/0)[x - (-4)]
x = -4
Equation of perpendicular bisector of BC:
y - (-5) = (-3/4)[x - (-4)]
y + 5 = (-3/4)x - 3
y = (-3/4)x - 8
Step 4: Find the intersection of perpendicular bisectors
We now find the point of intersection of the two perpendicular bisectors. Solving for x and y from the two equations, we get:
(-4, -8)
Therefore, the circumcenter of the triangle with vertices (-7, -1), (-1, -1), and (-7, -9) is (-4, -8).