Let's start by graphing the using the given coordinates.
To find the , we need equations for the of at least two sides of the triangle. Recall that a bisector cuts something in half, so we want perpendicular lines that perfectly divide the side of the triangle in half.
Finding Perpendicular Bisectors
By the , we know that and lines are .
Since DF is horizontal, any perpendicular line will be vertical. Examining the diagram, we can also identify the of this side.
Given the information, we know that the perpendicular bisector through DF has the equation x=0. To find the other perpendicular bisector, we need to find the equation of the line perpendicular to DE, passing through the midpoint between the D and E. This will be a four-step process.
- Find the slope of the line DE,
- Find the slope of the line perpendicular to DE,
- Find the midpoint between D and E,
- Find the equation of the perpendicular bisector.
Let's start with the first step.
Finding the Slope of DE
To find the slope of DE, let's use the .
m = y_2 - y_1/x_2 - x_1
The (x_1, y_1) and (x_2, y_2) are the coordinates of two points lying on DE. Since we know the coordinates of D and E, we can use them to find the slope.
m = y_2 - y_1/x_2 - x_1
m=6- 0/0- 6
m=6/-6
m=-6/6
m=-1
The slope of DE is -1.
Finding the Slope of the Line Perpendicular to DE
By The Slopes of Perpendicular Lines Theorem, the slope of any line perpendicular to DE will be the of the slope of DE, that is, -1.
m_1 m_2 = -1
-1(m_2) = -1
m_2 = -1/-1
m_2 = 1
The slope of the perpendicular line is 1. This allows us to write the following partial formula for the perpendicular bisector of DE.
y = 1x + b
To find the value of b we need to find the coordinates of one point from the bisector. Since the midpoint between D and E
must lie on the perpendicular bisector of DE, this will be the point we will find.
Finding the Midpoint Between D and E
To find the midpoint between D and E we will use the .
M( x_1 + x_2/2, y_1 + y_2/2 )
Let's substitute the coordinates of D and E into this formula.
M ( x_1 + x_2/2, y_1 + y_2/2 )
M( 6+ 0/2, 0+ 6/2)
M(6/2,6/2)
M(3,3)
The midpoint of DE is (3,3).
Finding the Equation of the Perpendicular Bisector
The last step in finding the equation of the perpendicular bisector is to substitute the found coordinates for x and y into the partial formula for the bisector to find the value of b.
Therefore, the equation of the perpendicular bisector of DE is y=x. Having found the equation of the perpendicular bisector DE, we can graph it on the same graph we used for the perpendicular bisector of EF. Note that this line should also pass through (0, 0), because b= 0 is the y-value of the of our line.
Finding the Circumcenter
The triangle's circumcenter is the point at which the perpendicular bisectors intersect.
The circumcenter is located at (0,0).
