How to find the circumcenter with coordinates?
Understand the Problem
The question is asking for the method to calculate the circumcenter of a triangle given the coordinates of its vertices. The circumcenter is the point where the perpendicular bisectors of the sides of the triangle meet, and it can be found using geometric and algebraic approaches.
Answer
Circumcenter coordinates are found at the intersection of the perpendicular bisectors of the triangle's sides.
Answer for screen readers
The circumcenter coordinates are ( (x, y) ) determined from the intersection of perpendicular bisectors.
Steps to Solve
- Identify the Triangle Vertices
Let's denote the triangle vertices as ( A(x_1, y_1) ), ( B(x_2, y_2) ), and ( C(x_3, y_3) ).
- Calculate Midpoints of Triangle Sides
Next, find the midpoints of the sides of the triangle.
For side ( AB ):
$$ M_{AB} = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) $$
For side ( BC ):
$$ M_{BC} = \left( \frac{x_2 + x_3}{2}, \frac{y_2 + y_3}{2} \right) $$
For side ( CA ):
$$ M_{CA} = \left( \frac{x_3 + x_1}{2}, \frac{y_3 + y_1}{2} \right) $$
- Calculate Slopes of Triangle Sides
Now, calculate the slopes of the sides ( AB ) and ( BC ).
The slope of line ( AB ) is given by:
$$ m_{AB} = \frac{y_2 - y_1}{x_2 - x_1} $$
And the slope of line ( BC ) is given by:
$$ m_{BC} = \frac{y_3 - y_2}{x_3 - x_2} $$
- Determine the Slopes of the Perpendicular Bisectors
The slopes of the perpendicular bisectors are the negative reciprocals of the slopes of the sides.
For the perpendicular bisector of ( AB ):
$$ m_{PB_{AB}} = -\frac{1}{m_{AB}} $$
For the perpendicular bisector of ( BC ):
$$ m_{PB_{BC}} = -\frac{1}{m_{BC}} $$
- Write the Equations of the Perpendicular Bisectors
Using the midpoint and slope, we can write the equations of the perpendicular bisectors:
For ( PB_{AB} ):
$$ y - M_{AB_y} = m_{PB_{AB}} (x - M_{AB_x}) $$
For ( PB_{BC} ):
$$ y - M_{BC_y} = m_{PB_{BC}} (x - M_{BC_x}) $$
- Find the Intersection Point of the Perpendicular Bisectors
To find the circumcenter, solve the equations of the two perpendicular bisectors simultaneously.
Set the equations equal to each other and solve for ( x ) and ( y ).
- Final Coordinates of the Circumcenter
The point ( (x, y) ) calculated from the system of equations gives the coordinates of the circumcenter of the triangle.
The circumcenter coordinates are ( (x, y) ) determined from the intersection of perpendicular bisectors.
More Information
The circumcenter is the center of the circumcircle, which is the circle passing through all three vertices of the triangle. The circumcenter can lie inside, on, or outside the triangle depending on the type of triangle (acute, right, or obtuse).
Tips
- Mix-up between the midpoints and the coordinates of the vertices.
- Incorrect calculation of slopes, which can lead to errors in the perpendicular slopes.
- Forgetting to check whether the division for slopes doesn't lead to a zero division (in the case of vertical lines).
