Find the coordinates of the circumcenter of...?

Question

right triangle MNO with verticies M (-4,0), N (0,5), and O (10, -3).

Please explain to me how I could solve for this...

Thanks in advance!

Answer 1

The right angle has its vertix at N. ( You should test the slopes of MN and ON) With this verified in a right triangle the circumcenter is midpoint of the hypothenuse .C is midpoint of OM (3,-3/2)

Answer 2

In general There are 2 ways to do it. One is set up x and y equidistant from the 3 points - that is

sqrt((x + 4)^2 + y^2) = sqrt(x^2 + (y - 5)^2) = sqrt((x - 10)^2 + (y + 3)^2).

This is kind of messy. The second way is a little simpler. You have 3 points. Choose any 2 of them and find the line equidistant between them. Then choose another 2 and find the line equidistant between them. Find the intersection, and you have a point equidistant from all 3.

But this case is even easier, if you know this is a right triangle. A right triangle has the hypotenuse as the diameter of its circle, so find the midpoint of the hypotenuse, and you have it.

In this case, it looks to me like MO is the hypotenuse, so the midpoint of MO is ((10 + -4)/2, (0 + -3)/2) = (3, -3/2)