We are given a triangle with a perimeter of 272 and the product of its sides equal to 314. What is the product of the inradius and circumradius?
We are given a triangle with a perimeter of 272 and the product of its sides equal to 314.
What is the product of the inradius and circumradius?
\(\text{Let inradius $r = \dfrac{2A}{a+b+c} $ } \\ \text{Let circumradius $ R =\dfrac{abc}{4A} $ } \\ a+b+c = 272 \\ abc = 314 \)
\(\begin{array}{|rcll|} \hline rR &=& \dfrac{2A}{(a+b+c)} * \dfrac{abc}{4A} \\\\ rR &=& \dfrac{abc}{2(a+b+c)} \\\\ rR &=& \dfrac{314}{2(272)} \\\\ rR &=& \dfrac{157}{272} \\\\ \mathbf{rR} &=& \mathbf{0.57720588235} \\ \hline \end{array}\)