A circle passes through the vertices of a triangle with side-lengths $7\tfrac{1}{2},10,12\tfrac{1}{2}.$ The radius of the circle is:
$\text{(A) } \frac{15}{4}\quad \text{(B) } 5\quad \text{(C) } \frac{25}{4}\quad \text{(D) } \frac{35}{4}\quad \text{(E) } \frac{15\sqrt{2}}{2}$
The triangle that goes through all the vertices of the triangle is the circumcircle of the triangle.
$(7\frac{1}{2})^{2}+10^{2}=(12\frac{1}{2})^{2}$, so the triangle is a right triangle.The radius of a circumcircle 
of a right triangle is half the hypotenuse. $\frac{1}{2}\cdot \frac{25}{2}=\frac{25}{4}\implies$ $\fbox{C}$