Let's start by graphing the triangle using the given coordinates.
To find the circumcenter, we need to find the intersection of the perpendicular bisectors 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 midpoints.
Finding Perpendicular Bisectors
Let's first find the midpoints of two sides. To do so, we can use the Midpoint Formula.
Side
Points
M(x_1+x_2/2,y_1+y_2/2)
Midpoint
HJ
( -10,7), ( -6,3)
L(-10+( -6)/2,7+ 3/2)
L(-8,5)
JK
( -6,3), ( -2,3)
M(-6+( -2)/2,3+ 3/2)
M(-4,3)
Let's add these midpoints to our graph.
According to the Slopes of Perpendicular Lines Theorem, the product of the slopes of perpendicular lines is -1. This allows us to find the slope of the perpendicular bisector from the slope of the side. Since we know the endpoints of HJ, we can find its slope using the Slope Formula.
m = y_2-y_1/x_2-x_1
Let's substitute the given endpoints H(-10,7) and J(-6,3).
We found that the slope of HJ is -1. Now we can find the slope of the perpendicular bisector from the product. Let's call it m_p.
-1* m_p = -1 ⇒ m_p = 1
The slope of the perpendicular bisector of HJ is 1.
y=1x+b ⇔ y=x+b
To complete its equation, we need the y-intercept. We can use the fact that the line passes through the midpoint L(-8,5).
Therefore, the equation of the perpendicular bisector of HJ is y=x+13. To find the second one, notice that JK is horizontal. This means that the perpendicular bisector of JK is a vertical line. Moreover, since it passes through M( -4,3), its equation is x= -4.
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.
x=-4 y=x+13
We can solve it by substituting x= -4 in the second equation.
