Respuesta :
Answer:
Circumcenter is [tex](-5,13)[/tex].
Step-by-step explanation:
Given:
The three vertices of triangle OVW are [tex]O(0,0),V(0,26),W(-10,0)[/tex].
Circumcenter is a point inside the triangle which is equidistant from each of the vertices of the triangle.
Let [tex]C(x,y)[/tex] be the circumcenter.
Distance between two points [tex](x_{1},y_{1})[/tex] and [tex](x_{2},y_{2})[/tex] is given as:
[tex]D=\sqrt{(x_{2}-x_{1})^{2}+(y_{2}-y_{1})^{2}}[/tex]
So, as per the definition of circumcenter, distance of point [tex]O(0,0)[/tex] from point [tex]C(x,y)[/tex] is equal to distance of point [tex]V(0,26)[/tex] from point [tex]C(x,y)[/tex].
So, OC = VC
or [tex](OC)^{2}=(VC)^{2}[/tex]
[tex](x-0)^{2}+(y-0)^{2}=(x-0)^{2}+(y-26)^{2}\\x^{2}+y^{2}=x^{2}+y^2+26^{2}-(2\times 26y)\\x^{2}+y^{2}-x^{2}-y^2=676-52y\\0=676-52y\\52y=676\\y=\frac{676}{52}=13[/tex].
Similarly, distance of point [tex]O(0,0)[/tex] from point [tex]C(x,y)[/tex] is equal to distance of point [tex]W(-10,0)[/tex] from point [tex]C(x,y)[/tex].
[tex]OC=WC[/tex] or
[tex](OC)^{2}=(WC)^{2}[/tex]
[tex](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=-\frac{100}{20}=-5[/tex].
Therefore, the circumcenter of the triangle with the given vertices is [tex](-5,13)[/tex].
