To find the circumcenter of triangle ABC, we need to find the point that is equidistant from the three vertices of the triangle.
Step 1: Find the midpoints of two sides of the triangle.
The midpoint of AB is:
((1+1)/2, (6+4)/2) = (1, 5)
The midpoint of AC is:
((1+5)/2, (6+4)/2) = (3, 5)
Step 2: Find the slopes of the two sides of the triangle.
The slope of AB is:
(6-4)/(1-1) = 0
The slope of AC is:
(6-4)/(1-5) = -1/2
Step 3: Find the perpendicular bisectors of the two sides.
The equation of the perpendicular bisector of AB passing through (1, 5) has a slope perpendicular to 0, which is undefined. Therefore, it is a vertical line passing through (1, 5), and its equation is:
x = 1
The equation of the perpendicular bisector of AC passing through (3, 5) has a slope perpendicular to -1/2, which is 2. Therefore, we can use the point-slope form to find its equation:
y - 5 = 2(x - 3)
Simplifying:
y - 5 = 2x - 6
y = 2x - 1
Step 4: Find the intersection of the perpendicular bisectors.
Solving the system of equations formed by the two perpendicular bisectors:
x = 1
y = 2x - 1
Substituting x = 1 into the second equation:
y = 2(1) - 1
y = 1
Therefore, the circumcenter of triangle ABC is (1, 1).
Find the circumcenter of ABC with A(1,6) B(1,4) and C (5,4)
3 answers
wrong
Apologies for the mistake. Let's redo the calculations correctly.
Step 1: Find the midpoints of two sides of the triangle.
The midpoint of AB is:
((1+1)/2, (6+4)/2) = (1, 5)
The midpoint of AC is:
((1+5)/2, (6+4)/2) = (3, 5)
Step 2: Find the slopes of the two sides of the triangle.
The slope of AB is:
(6-4)/(1-1) = undefined
The slope of AC is:
(6-4)/(1-5) = -1/2
Step 3: Find the perpendicular bisectors of the two sides.
The equation of the perpendicular bisector of AB passing through (1, 5) has a slope perpendicular to 0, which is undefined. Therefore, it is a vertical line passing through (1, 5), and its equation is:
x = 1
The equation of the perpendicular bisector of AC passing through (3, 5) has a slope perpendicular to -1/2, which is 2. Therefore, we can use the point-slope form to find its equation:
y - 5 = 2(x - 3)
Simplifying:
y - 5 = 2x - 6
y = 2x - 1
Step 4: Find the intersection of the perpendicular bisectors.
Solving the system of equations formed by the two perpendicular bisectors:
x = 1
y = 2x - 1
Substituting x = 1 into the second equation:
y = 2(1) - 1
y = 1
Therefore, the circumcenter of triangle ABC is (1, 1).
Step 1: Find the midpoints of two sides of the triangle.
The midpoint of AB is:
((1+1)/2, (6+4)/2) = (1, 5)
The midpoint of AC is:
((1+5)/2, (6+4)/2) = (3, 5)
Step 2: Find the slopes of the two sides of the triangle.
The slope of AB is:
(6-4)/(1-1) = undefined
The slope of AC is:
(6-4)/(1-5) = -1/2
Step 3: Find the perpendicular bisectors of the two sides.
The equation of the perpendicular bisector of AB passing through (1, 5) has a slope perpendicular to 0, which is undefined. Therefore, it is a vertical line passing through (1, 5), and its equation is:
x = 1
The equation of the perpendicular bisector of AC passing through (3, 5) has a slope perpendicular to -1/2, which is 2. Therefore, we can use the point-slope form to find its equation:
y - 5 = 2(x - 3)
Simplifying:
y - 5 = 2x - 6
y = 2x - 1
Step 4: Find the intersection of the perpendicular bisectors.
Solving the system of equations formed by the two perpendicular bisectors:
x = 1
y = 2x - 1
Substituting x = 1 into the second equation:
y = 2(1) - 1
y = 1
Therefore, the circumcenter of triangle ABC is (1, 1).