Help I need help to find the coordinates of the circumcenter of each triangle. Isosceles triangle CDE with vertices C(0, 6), D(0, –6), and E(12, 0)
the circumcenter is the intersection of the perpendicular bisectors of the sides.
The midpoint of CD = (0,0)
The slope of CD is -12/0 is undefined, so CD is vertical
The perpendicular bisector is horizontal.
You want the line through (0,0) with slope 0 is the line y=0.
The midpoint of DE = (6,-6)
The slope of DE is 6/12 = 1/2
You slope of the bisector is -2
The line through (6,-6) with slope -2 is
y+6 = -2(x-6), or y = -2x + 6
the two bisectors intersect at (0,3), which is the circumcenter.
as usual, check my math.
To find the coordinates of the circumcenter of a triangle, we can follow these steps:
1. Find the midpoints of two sides of the triangle.
2. Calculate the slopes of those sides.
3. Find the perpendicular bisectors of those sides.
4. Determine the intersection point of the perpendicular bisectors. This point will be the circumcenter.
Let's apply these steps to find the circumcenter of triangle CDE:
Step 1: Find the midpoints of sides CD and CE.
The midpoint formula is ( (x1 + x2) / 2, (y1 + y2) / 2 ).
Midpoint of CD:
(x1, y1) = (0, 6)
(x2, y2) = (0, -6)
Midpoint of CD = ( (0 + 0) / 2, (6 + -6) / 2 ) = (0, 0)
Midpoint of CE:
(x1, y1) = (0, 6)
(x2, y2) = (12, 0)
Midpoint of CE = ( (0 + 12) / 2, (6 + 0) / 2 ) = (6, 3)
Step 2: Calculate the slopes of sides CD and CE.
The slope formula is (y2 - y1) / (x2 - x1).
Slope of CD:
(x1, y1) = (0, 6)
(x2, y2) = (0, -6)
Slope of CD = ( -6 - 6 ) / ( 0 - 0 ) = undefined
Slope of CE:
(x1, y1) = (0, 6)
(x2, y2) = (12, 0)
Slope of CE = ( 0 - 6 ) / ( 12 - 0 ) = -1/2
Step 3: Find the perpendicular bisectors of sides CD and CE.
To find the perpendicular bisector, we need to find the negative reciprocal of the slopes calculated in step 2.
Perpendicular bisector of CD:
Slope of CD = undefined, so the perpendicular bisector of CD is a vertical line passing through the midpoint of CD, which is the x-coordinate of the midpoint: x = 0.
Perpendicular bisector of CE:
Slope of CE = -1/2, so the negative reciprocal is 2.
Using point-slope form, we can write the equation of the line passing through the midpoint of CE (6, 3) with slope 2:
y - y1 = m(x - x1)
y - 3 = 2(x - 6)
Simplifying,
y - 3 = 2x - 12
y = 2x - 9
Step 4: Determine the intersection point of the perpendicular bisectors.
To find the circumcenter, we need to find the intersection point of the perpendicular bisectors. This point is the circumcenter of the triangle.
Since the perpendicular bisector of side CD is a vertical line passing through x = 0, and the equation of the perpendicular bisector of side CE is y = 2x - 9, we can find the intersection by substituting x = 0 into the equation for the line y = 2x - 9:
y = 2(0) - 9
y = -9
Therefore, the circumcenter is (0, -9).
To summarize, the coordinates of the circumcenter of triangle CDE, given the vertices C(0, 6), D(0, -6), and E(12, 0), are (0, -9).