Question

Find the circumcenter of a triangle with vertices O(0, 0), V(0, 26), and W(−10,0).

Answer 1

Circumcenter is
`(-5,13)`.

Explanation:

Given:

The three vertices of triangle OVW are
`O(0,0),V(0,26),W(-10,0)`.

Circumcenter is a point inside the triangle which is equidistant from each of the vertices of the triangle.

Let
`C(x,y)` be the circumcenter.

Distance between two points
`(x_(1),y_(1))` and
`(x_(2),y_(2))` is given as:

`D=\sqrt{(x_(2)-x_(1))^(2)+(y_(2)-y_(1))^(2)}`

So, as per the definition of circumcenter, distance of point
`O(0,0)` from point
`C(x,y)` is equal to distance of point
`V(0,26)` from point
`C(x,y)`.

So, OC = VC

or
`(OC)^(2)=(VC)^(2)`

`(x-0)^(2)+(y-0)^(2)=(x-0)^(2)+(y-26)^(2)\\x^(2)+y^(2)=x^(2)+y^2+26^(2)-(2* 26y)\\x^(2)+y^(2)-x^(2)-y^2=676-52y\\0=676-52y\\52y=676\\y=(676)/(52)=13`.

Similarly, distance of point
`O(0,0)` from point
`C(x,y)` is equal to distance of point
`W(-10,0)` from point
`C(x,y)`.

`OC=WC` or

`(OC)^(2)=(WC)^(2)`

`(x-0)^(2)+(y-0)^(2)=(x-(-10))^(2)+(y-0)^(2)\\x^(2)+y^(2)=(x+10)^(2)+y^2\\ x^(2)+y^(2)=x^(2)+100+20x+y^2\\x^(2)+y^(2)-x^(2)-y^2=100+20x\\0=100+20x\\ 20x=-100\\x=-(100)/(20)=-5`.

Therefore, the circumcenter of the triangle with the given vertices is
`(-5,13)`.