Find the coordinates of the circumcenter of triangle DEF D(6,0) E(0,6) F(-6,0)
To find the circumcenter of triangle DEF, we can follow these steps:
Step 1: Find the midpoints of two sides of the triangle.
Let's find the midpoint of segment EF:
Midpoint of EF = [(x1 + x2)/2, (y1 + y2)/2]
= [(0 + (-6))/2, (6 + 0)/2]
= [-3, 3]
Step 2: Find the slope of the line passing through two sides of the triangle.
Let's find the slope between segments DE and EF.
Slope of DE = (y2 - y1)/(x2 - x1)
= (0 - 6)/(6 - 0)
= -6/6
= -1
Slope of EF = (y2 - y1)/(x2 - x1)
= (6 - 0)/(-6 - 0)
= 6/-6
= -1
Step 3: Find the slope of the perpendicular bisector of two sides.
The slope of the perpendicular bisector is the negative reciprocal of the slope of the original line.
Perpendicular bisector slope = -1/(-1)
= 1
Step 4: Find the equation of the perpendicular bisector.
Using the midpoint (-3, 3) and the slope of the perpendicular bisector (1), we can use the point-slope form of the equation of a line:
y - y1 = m(x - x1), where (x1, y1) is the midpoint and m is the slope.
y - 3 = 1(x + 3)
y - 3 = x + 3
y = x + 6
Step 5: Find the point of intersection between the perpendicular bisector and the third side of the triangle.
Let's find the intersection point of the line y = x + 6 and segment DF.
Substituting y = x + 6 into the equation of segment DF:
-6 = x - 6
x = 0
Substituting x = 0 into y = x + 6:
y = 0 + 6
y = 6
So, the coordinates of the circumcenter of triangle DEF are (0, 6).
To find the coordinates of the circumcenter of a triangle, you can follow these steps:
Step 1: Calculate the midpoints of two sides of the triangle.
- The midpoint of DE can be found by taking the average of the x-coordinates and the average of the y-coordinates of points D and E.
Midpoint DE (x1, y1) = ((6 + 0) / 2, (0 + 6) / 2)
= (3, 3)
- Similarly, the midpoint of EF can be found by taking the average of the x-coordinates and the average of the y-coordinates of points E and F.
Midpoint EF (x2, y2) = ((0 + (-6)) / 2, (6 + 0) / 2)
= (-3, 3)
Step 2: Calculate the slopes of the two lines perpendicular to the sides of the triangle.
- The slope of a line perpendicular to a line with slope m is -1/m.
- The slope of DE is (y2 - y1) / (x2 - x1) = (3 - 3) / (-3 - 3) = 0.
The slope of the line perpendicular to DE is -1/0, which is undefined.
- The slope of EF is (y3 - y2) / (x3 - x2) = (0 - 3) / (-6 - (-3)) = -3/3 = -1.
The slope of the line perpendicular to EF is -1/(-1) = 1.
Step 3: Calculate the equations of the perpendicular bisectors of two sides of the triangle.
- The equation of a line with slope m passing through point (x1, y1) is given by:
y - y1 = m(x - x1)
- The perpendicular bisector of DE passes through the midpoint (3, 3).
Using a slope of undefined, the equation becomes: x = 3.
- The perpendicular bisector of EF passes through the midpoint (-3, 3).
Using a slope of 1, the equation becomes: y - 3 = 1(x - (-3)) => y - 3 = x + 3 => y = x + 6.
Step 4: Solve the system of equations formed by the equations of the perpendicular bisectors.
- The two lines intersect at the circumcenter of the triangle.
Solving the equations x = 3 and y = x + 6 simultaneously, we get x = 3 and y = 9.
Therefore, the coordinates of the circumcenter of triangle DEF are (3, 9).