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Question 1181582: Given three lines 3x + 2y -16 = 0 …eq. 1, 2x - y + 1 = 0 …eq. 2 and x - 4y + 4 = 0 …eq. 3, let A be the point of intersection of lines …eq. 1 and …eq. 2, B be the point of intersection of lines …eq. 2 and …eq. 3, and C be the point of intersection of lines …eq. 3 and …eq. 1. Find the coordinates of circumcenter K of triangle ABC.
Answer by CPhill(1987)   (Show Source):
You can put this solution on YOUR website! Here's how to find the circumcenter of triangle ABC:
**1. Find the Coordinates of A:**
Solve equations 1 and 2 simultaneously:
* 3x + 2y = 16
* 2x - y = -1 => y = 2x + 1
Substitute y in the first equation:
* 3x + 2(2x + 1) = 16
* 3x + 4x + 2 = 16
* 7x = 14
* x = 2
Substitute x = 2 back into y = 2x + 1:
* y = 2(2) + 1 = 5
So, A = (2, 5)
**2. Find the Coordinates of B:**
Solve equations 2 and 3 simultaneously:
* 2x - y = -1
* x - 4y = -4 => x = 4y - 4
Substitute x in the second equation:
* 2(4y - 4) - y = -1
* 8y - 8 - y = -1
* 7y = 7
* y = 1
Substitute y = 1 back into x = 4y - 4:
* x = 4(1) - 4 = 0
So, B = (0, 1)
**3. Find the Coordinates of C:**
Solve equations 1 and 3 simultaneously:
* 3x + 2y = 16
* x - 4y = -4 => x = 4y - 4
Substitute x in the first equation:
* 3(4y - 4) + 2y = 16
* 12y - 12 + 2y = 16
* 14y = 28
* y = 2
Substitute y = 2 back into x = 4y - 4:
* x = 4(2) - 4 = 4
So, C = (4, 2)
**4. Find the Perpendicular Bisectors:**
The circumcenter K is the intersection of the perpendicular bisectors of the sides of the triangle. Let's find the equations of two of them.
* **Perpendicular bisector of AB:**
* Midpoint of AB = ((2+0)/2, (5+1)/2) = (1, 3)
* Slope of AB = (5-1)/(2-0) = 2
* Slope of the perpendicular bisector = -1/2
* Equation: y - 3 = (-1/2)(x - 1) => x + 2y = 7
* **Perpendicular bisector of BC:**
* Midpoint of BC = ((0+4)/2, (1+2)/2) = (2, 1.5)
* Slope of BC = (2-1)/(4-0) = 1/4
* Slope of the perpendicular bisector = -4
* Equation: y - 1.5 = -4(x - 2) => 4x + y = 9.5
**5. Find the Intersection of the Perpendicular Bisectors:**
Solve the equations of the perpendicular bisectors simultaneously:
* x + 2y = 7
* 4x + y = 9.5
Multiply the second equation by 2:
* x + 2y = 7
* 8x + 2y = 19
Subtract the first equation from the second:
* 7x = 12
* x = 12/7
Substitute x back into x + 2y = 7:
* 12/7 + 2y = 7
* 2y = 37/7
* y = 37/14
So, the circumcenter K = (12/7, 37/14) or approximately (1.71, 2.64)
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