Question

If ${\Delta }_1,{\Delta }_2$ are areas of incircle and circumcircle of a triangle with sides $3,4$ and $5$ then $\frac{{\Delta }_1}{{\Delta }_2}=$

• A. $\frac{16}{25}$
• B. $\frac{4}{25}$
• C. $\frac{9}{25}$
• D. $\frac{9}{16}$

Answer 1

The correct option is B $\frac{4}{25}$

$sidesoftriangle3,4,5$
$as{3}^2+4^2=5^2$
$Henceitisarightangledtriangle$
$\frac{a}{\sin A}=2R$
$\Rightarrow a=2R$
$R=\frac{a}{2}=\frac{5}{2}$
$r=\frac{\Delta }{s}=\frac{\frac{1}{2}\times 4\times 3}{\frac{5+4+3}{2}}=1$
$\text{Thus}\frac{{\text{area of inscribed circle s}}_1}{{\text{area of circumscribed circle s}}_2}=\frac{\pi r^2}{\pi R^2}=\frac{1^2}{{\left(\frac{5}{2}\right)}^2}=\frac{4}{25}$