Question

If in a $△ABC,AD,BE$ and $CF$ are the attitudes and $R$ is the circumradius, then the circumradius $R_1$ of $△DEF$ is

• A. $2R$
• B. $R$
• C. $\frac{R}{2}$
• D. None of these

Answer 1

The correct option is C $\frac{R}{2}$

In $△DEF$, we have $EF=a\cos A,DE=c\cos C,DF=b\cos B$

Let $R_1$ be the circumradius of $△DEF$

then, $R_1=\frac{\left(a\cos A\right)\left(b\cos B\right)\left(c\cos C\right)}{4\left(Areaof△DEF\right)}=\frac{abc\cos A\cos B\cos C}{4.\frac{1}{2}.DF.DE\sin \left(\angle EDF\right)}$

$=\frac{abc\cos A\cos B\cos C}{2.\left(b\cos B\right)\left(c\cos C\right)\sin ({180}^0-2A)}[\because \angle EDF={180}^0-2A]$

$=\frac{a\cos A}{2\sin 2A}=\frac{2R\sin A\cos A}{2\sin 2A}=\frac{R}{2}$

$\Rightarrow R_1=\frac{R}{2}$