Triangle XYZ has vertices at X(2, 5), Y(8, 5) and Z(8, 1). Find the coordinates of the circumcenter
To find the circumcenter of a triangle, we need to find the intersection of the perpendicular bisectors of the triangle's sides.
First, let's find the equations of the three sides of the triangle.
The equation of the line passing through points X(2, 5) and Y(8, 5) can be found using the slope-intercept form:
Slope, m = (change in y-coordinate) / (change in x-coordinate) = (5 - 5) / (8 - 2) = 0 / 6 = 0
Since the line is horizontal, the equation becomes y = 5.
Similarly, the equation of the line passing through points Y(8, 5) and Z(8, 1) can be found using the slope-intercept form:
Slope, m = (change in y-coordinate) / (change in x-coordinate) = (1 - 5) / (8 - 8) = -4 / 0
Since the line is vertical, the equation becomes x = 8.
Now, let's find the equation of the line passing through points X(2, 5) and Z(8, 1) using the point-slope form:
Slope, m = (change in y-coordinate) / (change in x-coordinate) = (1 - 5) / (8 - 2) = -4 / 6 = -2/3
Using the point-slope form, we get:
y - 5 = (-2/3)(x - 2)
3y - 15 = -2x + 4
2x + 3y = 19
Now, let's find the midpoints of the three sides of the triangle.
The midpoint of side XY is ((2 + 8) / 2, (5 + 5) / 2) = (5, 5).
The midpoint of side YZ is ((8 + 8) / 2, (5 + 1) / 2) = (8, 3).
The midpoint of side XZ is ((2 + 8) / 2, (5 + 1) / 2) = (5, 3).
Now, let's find the slopes of the perpendicular bisectors of the three sides: (negative reciprocal of the slopes of the sides).
The slope of the perpendicular bisector of side XY is -1/0 = undefined (vertical line).
The slope of the perpendicular bisector of side YZ is -1/(-2/3) = 3/2.
The slope of the perpendicular bisector of side XZ is -1/(-2/3) = 3/2.
Now, let's use the point-slope form to find the equations of the perpendicular bisectors.
For the perpendicular bisector of side YZ:
Using the midpoint (8, 3) and the slope 3/2, we get:
y - 3 = (3/2)(x - 8)
2y - 6 = 3x - 24
3x - 2y = 18
For the perpendicular bisector of side XZ:
Using the midpoint (5, 3) and the slope 3/2, we get:
y - 3 = (3/2)(x - 5)
2y - 6 = 3x - 15
3x - 2y = 9
Now, let's find the intersection point of the perpendicular bisectors, which is the circumcenter of the triangle.
Let's solve the system of equations formed by the perpendicular bisectors:
3x - 2y = 18 ... (1)
3x - 2y = 9 ... (2)
Subtracting equation (2) from equation (1), we get:
(3x - 2y) - (3x - 2y) = 18 - 9
0 = 9
Since the equations are identical, we have infinitely many solutions. This implies that the circumcenter of the triangle does not exist.
Therefore, the triangle XYZ doesn't have a circumcenter.