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

Answer:

option A (-1, 2.5) is correct

Step-by-step 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

$OA = OB$

$OA^{2}=OB^{2}$

$(x-1)^{2}+(y-1)^{2} =(x-1)^{2}+(y-1)^{4}$

$x^{2} +1-2x+y^{2} +1-2y=x^{2} +1-2x+y^{2} +16-8y$

$8y-2y=16-1$

$y=\frac{15}{6}$

$y=2.5$

$OA = OC$

$OA^{2}=OC^{2}$

$(x-1)^{2}+(y-1)^{2} =(x-(-3))^{2}+(y-4)^{4}$

$(x-1)^{2}+(y-1)^{2} =(x+3)^{2}+(y-4)^{4}$

$x^{2} +1-2x+y^{2} +1-2y=x^{2} +9+6x+y^{2} +16-8y$

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

put the y value in this equation

$-8x+6\times\frac{15}{6} = 23$

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

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