find the center of the circle that you can circumscribe about ABC
A(2,8)
B(0,8)
C(2,2)
To find the center of the circle that can circumscribe about triangle ABC, you need to find the intersection point of the perpendicular bisectors of the triangle's sides.
First, let's find the equations of the perpendicular bisectors of the triangle's sides:
1. Perpendicular bisector of AB:
- Midpoint of AB: [(2+0)/2, (8+8)/2] = (1, 8)
- Slope of AB: (8-8)/(2-0) = 0
The equation of the perpendicular bisector of AB is x = 1.
2. Perpendicular bisector of BC:
- Midpoint of BC: [(2+2)/2, (8+2)/2] = (2, 5)
- Slope of BC: (2-2)/(2-2) = undefined
The equation of the perpendicular bisector of BC is y = 5.
3. Perpendicular bisector of AC:
- Midpoint of AC: [(2+2)/2, (8+2)/2] = (2, 5)
- Slope of AC: (5-8)/(2-2) = undefined
The equation of the perpendicular bisector of AC is y = 5.
Since the equation of the perpendicular bisector of BC is vertical (y = 5) and the equation of the perpendicular bisector of AB is horizontal (x = 1), the two lines intersect at the point (1, 5). Therefore, the center of the circumscribed circle is (1, 5).