Question

The radius of circumcircle of triangle PRS with three sides $2\sqrt{3},6\sqrt{2},10$ is
A. $5$
B. $3\sqrt{3}$
C. $3\sqrt{2}$
D. $2\sqrt{3}$

Answer 1

Hint: We have to know the cosine rule in a triangle that is $\cos P=\dfrac{{{r}^{2}}+{{s}^{2}}-{{p}^{2}}}{2rs}$ and we will get $\cos P$ and then find $\sin P$by trigonometric identity that is ${{\sin }^{2}}P+{{\cos }^{2}}P=1$ and then we have to know the sine rule in a triangle that is $\Delta PRS:\dfrac{p}{\sin P}=\dfrac{r}{\sin R}=\dfrac{s}{\sin S}=2R$. By this way we will get the circumradius of the triangle.

Complete step-by-step solution -
__
Given, the sides of triangle PRS are $2\sqrt{3},6\sqrt{2},10$
Hence p= $2\sqrt{3}$. . . . . . . . . . . . . . . . . . . . . . . (1)
r= $6\sqrt{2}$. . . . . . . . . . . . . . . . . . . . . . . .(2)
s= $10$. . . . . . . . . . . . . . . . . . . . . . . . . (3)
we know that in a triangle according to cosine rule $\cos P=\dfrac{{{r}^{2}}+{{s}^{2}}-{{p}^{2}}}{2rs}$
$\cos P=\dfrac{{{10}^{2}}+{{\left( 6\sqrt{2} \right)}^{2}}-{{\left( 2\sqrt{3} \right)}^{2}}}{2\times 10\times 6\sqrt{2}}$
$\cos P=\dfrac{100+72-12}{120\sqrt{2}}$
$\cos P=\dfrac{160}{120\sqrt{2}}=\dfrac{4}{3\sqrt{2}}$
Rationalizing the both numerator and denominator by multiplying with $\sqrt{2}$ we will get,
$\cos P=\dfrac{4}{3\sqrt{2}}\times \dfrac{\sqrt{2}}{\sqrt{2}}=\dfrac{2\sqrt{2}}{3}$. . . . . .. . . . . . . .(4)
We know the trigonometric identity that ${{\sin }^{2}}P+{{\cos }^{2}}P=1$
By substituting the value of $\cos P$ we will get $\sin P$
$\sin P=\sqrt{1-\dfrac{8}{9}}=\sqrt{\dfrac{1}{9}}$
$\sin P=\dfrac{1}{3}$. . . . . . . . .. . . .(5)
According to sine rule we know that in a triangle, $\Delta PRS:\dfrac{p}{\sin P}=\dfrac{r}{\sin R}=\dfrac{s}{\sin S}=2R$
We can take the term $\dfrac{p}{\sin P}=2R$
By substituting the values of p and $\sin P$ we will get the circumradius of triangle PRS as follows
$\dfrac{2\sqrt{3}}{\dfrac{1}{3}}=2R$
$2R=6\sqrt{3}$
$R=3\sqrt{3}$
So, the correct option is option (B).

Note: In the above we find $\cos P$ by using cosine rule it is not mandatory to find only $\cos P$ we can find $\cos R$ or else $\cos S$ and then using the trigonometric identity find $\sin R$ and $\sin S$, by applying the sine rule we can find the circumradius of triangle PRS.