Practice Problem 5
Lynn Palmer
Given that point B is at (4,5), point R is at (1,0) and point K is at (6,1), find the approximate circumcenter of triangle BRK. Is the distance from the circumcenter to each of the vertices equal?
The circumcenter of a triangle is the point of concurrency of the three perpendicular bisectors in a triangle. So, first, we need to find the perpendicular bisectors of each segment. For segment BR, we can use the midpoint formula to solve for the midpoint.
((4+1)/2 , (5+0)/2)
(5/2 , 5/2)
(2.5 , 2.5)
Mark the midpoint on the graph. Next, we need to find the line that is perpendicular to segment BR. To find this, we need to find the opposite reciprocal slope of segment BR. We can use the slope formula to find the slope.
((0-5)/(1-4))
(-5/-3)
(5/3)
After that, we need to find the opposite reciprocal of 5/3, which is -3/5. Construct the line which has a slope of -3/5 and passes through the midpoint of segment BR, which is (2.5,2.5).
Repeat the process to find the perpendicular bisectors for segment RK.
Segment RK:
Midpoint: ((1+6)/2 , (0+1)/2)
(7/2,1/2)
(3.5,0.5)
Slope: (1-0)/6-1)
1/5
Opposite reciprocal: -5/1
Construct the line that passes through the midpoint (3.5 , 0.5)
Now that we have the two lines drawn, we can approximate the circumcenter of triangle BRK. The intersection of the perpendicular bisector of RK and the perpendicular bisector of BR is extremely close to the point (3,2). From this, we can approximate that the circumcenter of triangle BRK is (3,2).
The circumcenter of a triangle is the point of concurrency of the three perpendicular bisectors in a triangle. So, first, we need to find the perpendicular bisectors of each segment. For segment BR, we can use the midpoint formula to solve for the midpoint.
((4+1)/2 , (5+0)/2)
(5/2 , 5/2)
(2.5 , 2.5)
Mark the midpoint on the graph. Next, we need to find the line that is perpendicular to segment BR. To find this, we need to find the opposite reciprocal slope of segment BR. We can use the slope formula to find the slope.
((0-5)/(1-4))
(-5/-3)
(5/3)
After that, we need to find the opposite reciprocal of 5/3, which is -3/5. Construct the line which has a slope of -3/5 and passes through the midpoint of segment BR, which is (2.5,2.5).
Repeat the process to find the perpendicular bisectors for segment RK.
Segment RK:
Midpoint: ((1+6)/2 , (0+1)/2)
(7/2,1/2)
(3.5,0.5)
Slope: (1-0)/6-1)
1/5
Opposite reciprocal: -5/1
Construct the line that passes through the midpoint (3.5 , 0.5)
Now that we have the two lines drawn, we can approximate the circumcenter of triangle BRK. The intersection of the perpendicular bisector of RK and the perpendicular bisector of BR is extremely close to the point (3,2). From this, we can approximate that the circumcenter of triangle BRK is (3,2).