Mathematics - Properties of Triangle Question with Solution | TestHub

First, calculate the semi-perimeter $s=\frac{a+b+c}{2}=\frac{13+14+15}{2}=\frac{42}{2}=21$. The area of the triangle $\Delta$ can be found using Heron's formula: $\Delta =\sqrt{s(s-a)(s-b)(s-c)}=\sqrt{21(21-13)(21-14)(21-15)}=\sqrt{21\cdot 8\cdot 7\cdot 6}=\sqrt{7056}=84$. The circumradius $R$ is given by $R=\frac{abc}{4\Delta }$. Substituting the values, $R=\frac{13\cdot 14\cdot 15}{4\cdot 84}=\frac{2730}{336}$. Simplifying the fraction, $R=\frac{65}{8}$.