Question

A triangle has vertices at (1, 1), (1, 4), and (-3, 4).

What are the coordinates of the circumcenter?​

A. (-1, 2.5)
B (-1/3, 3)
C (0, 3)
D (1, 4)

Answer 1

option A (-1, 2.5) is correct

Explanation:

Given

Let A(1, 1)

Let B(1, 4)

Let C(-3, 4)

Let coordinates of the circumcenter is (x, y)

Using distance formula

`OA = OB`

`OA^(2)=OB^(2)`

`(x-1)^(2)+(y-1)^(2) =(x-1)^(2)+(y-1)^(4)`

`x^(2) +1-2x+y^(2) +1-2y=x^(2) +1-2x+y^(2) +16-8y`

`8y-2y=16-1`

`y=(15)/(6)`

`y=2.5`

`OA = OC`

`OA^(2)=OC^(2)`

`(x-1)^(2)+(y-1)^(2) =(x-(-3))^(2)+(y-4)^(4)`

`(x-1)^(2)+(y-1)^(2) =(x+3)^(2)+(y-4)^(4)`

`x^(2) +1-2x+y^(2) +1-2y=x^(2) +9+6x+y^(2) +16-8y`

`-6x-2y-2x+8y=25-2`

put the y value in this equation

`-8x+6*(15)/(6) = 23`

`-8x=23-15\\-8x=8\\x=-1`

So the coordinates of the circumcenter is (1, 2.5)