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:option A (-1, 2.5) is correctStep-by-step explanation:GivenLet A(1, 1)Let B(1, 4)Let C(-3, 4)Let coordinates of the circumcenter is (x, y)Using distance formula[tex]OA = OB[/tex][tex]OA^{2}=OB^{2}[/tex][tex](x-1)^{2}+(y-1)^{2} =(x-1)^{2}+(y-1)^{4}[/tex][tex]x^{2} +1-2x+y^{2} +1-2y=x^{2} +1-2x+y^{2} +16-8y[/tex][tex]8y-2y=16-1[/tex][tex]y=\frac{15}{6}[/tex][tex]y=2.5[/tex][tex]OA = OC[/tex][tex]OA^{2}=OC^{2}[/tex][tex](x-1)^{2}+(y-1)^{2} =(x-(-3))^{2}+(y-4)^{4}[/tex][tex](x-1)^{2}+(y-1)^{2} =(x+3)^{2}+(y-4)^{4}[/tex][tex]x^{2} +1-2x+y^{2} +1-2y=x^{2} +9+6x+y^{2} +16-8y[/tex][tex]-6x-2y-2x+8y=25-2[/tex]put the y value in this equation[tex]-8x+6\times\frac{15}{6} = 23[/tex][tex]-8x=23-15\\-8x=8\\x=-1[/tex]So the coordinates of the circumcenter is (1, 2.5)