Find the coordinates of the circumcenter of the triangle with the given vertices. X(-2, 1), Y(2, -3), Z(6, -3).

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14-01-2020
Mathematics

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Find the coordinates of the circumcenter of the triangle with the given vertices. X(-2, 1), Y(2, -3), Z(6, -3).

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hondeashok12 hondeashok12 · Answer 1 · 2020-01-15T10:09:50+01:00

Answer:

C (4,3)

Step-by-step explanation:

Steps to find the circumcenter:

1) Find and Calculate the midpoint of given coordinates or midpoints.

2) Calculate the slope of the perpendicular bisector lines

3) By using the midpoint and the slope, find out the equation of the line

[tex]( y-y_1 )= m(x-x_1)[/tex]

4) Do the same for another line

Solve the equations of two lines to get the intersection point.

The intersection point will be the circumcenter of the triangle.

Given :

X(-2,1), Y(2,-3), Z(6,-3)

Please see the attached file:

MrRoyal MrRoyal · Answer 2 · 2022-01-29T09:15:59+01:00

The coordinates of the circumcenter of the triangle with the vertices (-2, 1), (2, -3) and (6, -3) is (4,3)

The coordinates of the triangle are given as:

X = (-2, 1)

Y = (2, -3)

Z = (6, -3)

Assume the coordinate of the circumcenter (C) is:

C = (x,y)

Calculate the distance between point C and each of the coordinates using the distance formula.

So, we have:

[tex]cx = \sqrt{(x + 2)^2 + (y - 1)^2}[/tex]

[tex]cy = \sqrt{(x - 2)^2 + (y + 3)^2}[/tex]

[tex]cz = \sqrt{(x - 6)^2 + (y + 3)^2}[/tex]

Equate the above equations

[tex]\sqrt{(x + 2)^2 + (y - 1)^2} = \sqrt{(x - 2)^2 + (y + 3)^2} = \sqrt{(x - 6)^2 + (y + 3)^2}[/tex]

Square all three sides

[tex](x + 2)^2 + (y - 1)^2 = (x - 2)^2 + (y + 3)^2 = (x - 6)^2 + (y + 3)^2[/tex]

Rewrite as:

[tex](x + 2)^2 + (y - 1)^2 = (x - 2)^2 + (y + 3)^2[/tex]

[tex](x + 2)^2 + (y - 1)^2 = (x - 6)^2 + (y + 3)^2[/tex]

Expand the equations

[tex]x^2 + 4x + 4 + y^2 - 2y + 1 = x^2 -4x + 4 + y^2 + 6y + 9[/tex]

[tex]x^2 + 4x + 4 + y^2 - 2y + 1 = x^2 -12x + 36 + y^2 + 6y + 9[/tex]

Simplify the equations

[tex]4x - 2y + 1 = -4x + 6y + 9[/tex]

[tex]4x + 4 - 2y + 1 =-12x + 36 + 6y + 9[/tex]

Further, simplify the equations

[tex]4x+4x - 2y -6y = -1 + 9[/tex]

[tex]4x + 12x - 2y - 6y = 36 + 9-4-1[/tex]

Further, simplify the equations

[tex]8x - 8y = 8[/tex]

[tex]16x - 8y = 40[/tex]

Subtract the second equation, from the first

[tex]8x - 16x -8y + 8y = 8 -40[/tex]

[tex]-8x = -32[/tex]

Solve for x

[tex]x = 4[/tex]

Substitute 4 for x in

[tex]8(4) - 8y = 8[/tex]

[tex]32 - 8y = 8[/tex]

Collect like terms

[tex]- 8y = 8 -32[/tex]

[tex]- 8y = -24[/tex]

Divide both sides by -8

[tex]y = 3[/tex]

Hence, the coordinates of the circumcenter is (4,3)