Let's start by graphing the triangle using the given coordinates.
To find the , we need to find the intersection of the of the sides of the triangle. This means that we need to know the equations of at least two of them. Recall that a bisector cuts something in half, so we want to find lines that are perpendicular to the sides at their .
Finding Perpendicular Bisectors
Since we can find perpendicular bisectors of any of two sides, let's do it for AB and AC. Let's first find their midpoints. To do so, we can use the .
| Side
|
Points
|
M(x_1+x_2/2,y_1+y_2/2)
|
Midpoint
|
| AB
|
( 2,5), ( 6,6)
|
D(2+ 6/2,5+ 6/2)
|
D(4,11/2)
|
| AC
|
( 2,5), ( 12,3)
|
E(2+ 12/2,5+ 3/2)
|
E(7,4)
|
Let's add these midpoints to our graph.
Once we know the midpoints, we are ready to find the equations of our perpendicular bisectors.
Perpendicular Bisector of AB
According to the , the product of the slopes of lines is -1. Let's find the slope of AB first. Since we know the endpoints of AB, we can find its slope using the .
m = y_2-y_1/x_2-x_1
Let's substitute the given endpoints A( 2,5) and B( 6,6).
m_(AB) = y_2-y_1/x_2-x_1
m_(AB)=6- 5/6- 2
m_(AB)=1/4
We found that the slope of AB is 14. Now we can find the slope of the perpendicular bisector from the product. Let's call it m_(p_1).
1/4* m_(p_1) = -1 ⇔ m_(p_1) = -4
The slope of the perpendicular bisector of AB is -4.
y=-4x+b
To complete its equation, we need the y-intercept. We can use the fact that the line passes through the midpoint D( 4, 112).
y=-4 x+b
11/2=-4( 4)+b
11/2=-16+b
11/2+16=b
11/2+32/2=b
43/2=b
b=43/2
Therefore, the equation of the perpendicular bisector of AB is y=-4 x+ 432.
Perpendicular Bisector of AC
Similarly, we can find the equation of the perpendicular bisector of AC. Let's first find the slope of AC using the Slope Formula and the given end points A( 2, 5) and C( 12,3).
m_(AC) = y_2-y_1/x_2-x_1
m_(AC)=3- 5/12- 2
m_(AC)=-1/5
Once we know the slope of AC, we can find the slope of its perpendicular bisector from the product. Let's call it m_(p_2)
-1/5* m_(p_2) = -1 ⇔ m_(p_2) = 5
The slope of the perpendicular bisector of AC is 5.
y=5x+b
Once again, to complete its equation, we can substitute the coordinates of its midpoint E(7,4).
Therefore, the equation of the perpendicular bisector of AC is y=5x-31.
Finding the Circumcenter
We want to find the intersection of the two perpendicular bisectors that we found. To do it, we need to solve the system of their equations.
y=-4 x+43/2 y=5x-31
We can solve it by substituting y=-4 x+ 432 in the second equation.
y=-4 x+43/2 & (I) y=5x-31 & (II)
y=-4 x+43/2 -4 x+43/2=5x-31
y=-4 x+43/2 43/2=9x-31
y=-4 x+43/2 31+43/2=9x
y=-4 x+43/2 62/2+43/2=9x
y=-4 x+43/2 105/2=9x
y=-4 x+43/2 105/18=x
y=-4 x+43/2 35/6=x
y=-4 x+43/2 x=35/6
We found that x= 356. Now we can substitute it to the first equation to find the value of y.
y=-4 x+43/2 x=35/6
y=-4( 35/6)+43/2 x=35/6
y=- 1406+ 432 x= 356
y=- 1406+ 1296 x= 356
y=- 116 x= 356
Therefore, the coordinates of the circumcenter are ( 356,- 116).
