Question
The equations of the three sides of a triangle are $$x=2,\, y+1=0$$ and $$x+2y=4.$$ The coordinates of the circumcenter of the triangle are :
A.
$$\left( {4,\, 0} \right)$$
B.
$$\left( {2,\, - 1} \right)$$
C.
$$\left( {0,\, 4} \right)$$
D.
none of these
Answer :
$$\left( {4,\, 0} \right)$$
Solution :
One side of the triangle is parallel to the $$y$$-axis and another side is parallel to the $$x$$-axis. So, the triangle is a right-angled triangle. Hence, the middle point of the hypotenuse is the circumcenter. Solving $$x=2,\,x+2y=4$$ we get one end of the hypotenuse, and solving $$y+1=0,\,x+2y=4$$ we get the other end. Their coordinates are $$\left( {2,\,1} \right)$$ and $$\left( {6,\, - 1} \right).$$
$$\therefore {\text{circumcentre}} = \left( {\frac{{2 + 6}}{2},\,\frac{{1 - 1}}{2}} \right)$$