Question

The incircle of a scalene triangle $ABC$ touches side $AB,BC$ and $AC$ at $P,Q$ and $R$ respectively. If $O$ is the incentre of triangle $ABC,$ then which of the following statements is (are) correct?

$\text{I.}$ Area of $△OBC$ is $\frac{1}{2}\times BC\times \text{radius of incircle}$
$\text{II.}$ $AB+CQ=BQ+AC$
$\text{III.}$ Area of $△ABC$ is $\frac{1}{2}\times \text{perimeter of}△ABC\times \text{radius of incircle}$

• A. $\text{I}$ and $\text{III}$
• B. $\text{I}$ only
• C. $\text{II}$ and $\text{III}$
• D. $\text{I},\text{II}$ and $\text{III}$

Answer 1

The correct option is D $\text{I},\text{II}$ and $\text{III}$



$\text{I}.$ Area of $△OBC=\frac{1}{2}\times OQ\times BC$

$\text{II}.$ $AB+CQ=x+y+z$
$BQ+AC=y+x+z\Rightarrow AB+CQ=BQ+AC$

$\text{III}.$ Area of $△ABC=ar\left(△OBC\right)+ar\left(△OAC\right)+ar\left(△OAB\right)$
$=\frac{1}{2}\times BC\times r+\frac{1}{2}\times AC\times r+\frac{1}{2}\times AB\times r$
$=\frac{1}{2}(BC+AC+AB)r$
$=\frac{\text{perimeter}\cdot \text{inradius}}{2}$