Find the circumcenter of a triangle with the given vertices. #1) H(5,0), J(0,3), K(0,0) #2) L(0,0), M(-2, 0), N(0, -4)

1)

We see that if we plot the
coordinates on the graph,  the triangle HKJ is a right angled triangle , with right angle at
 K(0,0)  and  HJ as hypotenuse. Therefore the mid point of the hypotenuse HJ is the circumcentre
. The coordinates of  H and J are : H (5,0) and J(0,3).

Therefore
the mid point of HJ  is given by :

Mx = (Hx+Jx)/2 = ((5+0)/2 =
2.5. 

My = (Hy + Jy)/2 = (0+3)/2 =
1.5.

Therefore the circumcentre of the triangle HKJ is  (2.5 ,
1.5).

2)

The triangle MLN is a right
angled triangle with right angle at L(0,0). The hypotenuse of the triangle is MN with M at (-2,0)
and N at (0,4).

So the mid point of the hypotenuse MN is the
circumcentre.

The mid point C of MN  has the coordinates given
by:

Cx = (Mx+Nx)/2 = ((-2+0)/2 = -1.

Cy
= (My+Ny)/2 = ((0-4)/2 = =-2.

Therefore the circumcentre of the
triangle MLN is  (-1,-2).