The circumcenter is given by the point of concurrency of the
perpendicular bisectors to the sides AB and AC of ΔABC.
Correct response:
- The coordinates of the circumcenter is [tex]\underline{(3, \, 1.5)}[/tex]
Method used to find the circumcenter of the triangle
The given vertices of the triangle ΔABC are; A(4, 6), B(6, -2), and C(0, -2)
Required:
The coordinates of the circumcenter of ΔABC
Solution:
The circumcenter is the point of concurrency of the perpendicular
bisectors of the sides of the triangle, which is found as follows;
[tex]Slope \ of \ side \ AB = \mathbf{\dfrac{-2 - 6}{6 - 4}} = -4[/tex]
- [tex]Coordinates \ of \ the \ midpoint \ of \ AB = \left(\dfrac{4 + 6}{2} , \ \dfrac{6 + (-2)}{2} \right) = \mathbf{(5, \ 2)}[/tex]
Equation of the perpendicular line to AB is therefore;
[tex](y - 2) = \mathbf{\dfrac{1}{4} \cdot (x - 5)}[/tex]
Which gives;
4·y - 8 = x - 5
x = 4·y - 8 + 5 = 4·y - 3
x = 4·y - 3
[tex]Slope \ of \ side \ AC = \dfrac{-2 - 6}{0 - 4} = \mathbf{2}[/tex]
- [tex]Coordinates \ of \ midpoint \ of \ AC = \left(\dfrac{4 + 0}{2} , \ \dfrac{6 + (-2)}{2} \right) = \mathbf{(2, \, 2)}[/tex]
Equation of the perpendicular line to AC is therefore;
[tex]y - 2 = \mathbf{ -\dfrac{1}{2} \cdot (x - 2)}[/tex]
Which gives;
4 - 2·y = x - 2
Therefore;
x = 4 - 2·y + 2 = 6 - 2·y
Equating both values of x gives;
4·y - 3 = 6 - 2·y
6·y = 6 + 3 = 9
Therefore;
[tex]y = \dfrac{9}{6} = \dfrac{3}{2} = \mathbf{1.5}[/tex]
The value of y at the circumcenter is 1.5
Therefore;
At the circumcenter, we have;
x = 6 - 2 × 1.5 = 3
- The coordinates of the circumcenter is [tex]\underline{(3, \, 1.5)}[/tex]
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