Find the center of the circle that can be circumscribed about the triangle.
(2, 0)
(0, 1)
(0, 0)
(–1, 3)
To find the center of the circle circumscribed about a triangle, we first need to find the circumcenter, which is the point where the perpendicular bisectors of the sides of the triangle intersect.
Given the three vertices of the triangle are (2, 0), (0, 1), and (-1, 3).
To find the circumcenter, we need to find the equations of the perpendicular bisectors of the sides of the triangle. Then we can find the point of intersection of those bisectors, which will be the circumcenter.
Let's find the equations of the perpendicular bisectors of the sides of the triangle:
1. Perpendicular bisector of the line passing through (2, 0) and (0, 1):
- Midpoint of the line segment joining (2, 0) and (0, 1) = ((2+0)/2, (0+1)/2) = (1, 1/2)
- Slope of the line passing through (2, 0) and (0, 1) = (1-0)/(0-2) = -1/2
- Slope of the perpendicular bisector = 2 (negative reciprocal of -1/2)
- Equation of the perpendicular bisector passing through (1, 1/2) = y - 1/2 = 2(x - 1)
2. Perpendicular bisector of the line passing through (2, 0) and (-1, 3):
- Midpoint of the line segment joining (2, 0) and (-1, 3) = ((2+(-1))/2, (0+3)/2) = (1/2, 3/2)
- Slope of the line passing through (2, 0) and (-1, 3) = (3-0)/(-1-2) = -1
- Slope of the perpendicular bisector = 1 (negative reciprocal of -1)
- Equation of the perpendicular bisector passing through (1/2, 3/2) = y - 3/2 = 1(x - 1/2)
Solving these two equations, we get the circumcenter to be (3/2, 3/2). Therefore, the center of the circle that can be circumscribed about the triangle is (3/2, 3/2).
Are you sure
Let's recheck the calculations for the circumcenter of the triangle given by the vertices (2, 0), (0, 1), and (-1, 3).
First, we find the midpoints of the sides of the triangle:
1. Midpoint of the line segment joining (2, 0) and (0, 1):
Midpoint = ((2 + 0)/2, (0 + 1)/2) = (1, 1/2)
2. Midpoint of the line segment joining (2, 0) and (-1, 3):
Midpoint = ((2 + (-1))/2, (0 + 3)/2) = (1/2, 3/2)
3. Midpoint of the line segment joining (0, 1) and (-1, 3):
Midpoint = ((0 + (-1))/2, (1 + 3)/2) = (-1/2, 2)
Next, we find the slopes of the lines passing through the midpoints and perpendicular to the corresponding sides:
1. Slope of the line passing through (1, 1/2) and perpendicular to the line connecting (2, 0) and (0, 1):
Slope = 1
2. Slope of the line passing through (1/2, 3/2) and perpendicular to the line connecting (2, 0) and (-1, 3):
Slope = -1/2
Using the two midpoints and their corresponding slope, we find the intersection point to get the circumcenter.
Equation of the line passing through (1, 1/2) with slope 1:
y - 1/2 = 1(x - 1)
y = x - 1/2
Equation of the line passing through (1/2, 3/2) with slope -1/2:
y - 3/2 = -1/2(x - 1/2)
y = -x + 5/2
Solving the two equations gives us the coordinates of the circumcenter as (3, 2). Therefore, the center of the circle that can be circumscribed about the given triangle is (3, 2). Thank you for pointing out that error, and I appreciate your attention to detail.