Question

Statement 1 : Let the vertices of a $△ABC$ are $A(-5,-2),B(7,6)$ and $C(5,-4)$. Then coordinates of circumcenter are $(1,2)$.

Statement 2 : In a right-angle triangle, midpoint of hypotenuse is the circumcenter of the triangle.

Answer 1

We have,
$\left(m_{AC}\right)\left(m_{BC}\right)=\left(\frac{-4+2}{5+5}\right)\left(\frac{-4-6}{5-7}\right)=-1$
Therefore, $△ABC$ is a right-angled triangle with $C$ as the right angle.
Hence, circumcenter is midpoint of $AB$, i.e., $(1,2)$.

Hence, Statement 1 is true, statement 2 is true;

Statement 2 is correct explanation for satement 1.