Question

Let $P$ be an interior point in an acute angled $△ABC.$ Let $x,y,z$ be the lengths of the perpendicular from the point $P$ to the sides $BC,CA\text{and}AB$ respectively. If $AB=c,BC=a\text{and}CA=b,$ then the value of $\frac{1}{2}(ax+by+cz)$

• A. is always greater than the area of $△ABC$
• B. is equal to the area of $△ABC$
• C. is less than the area of $△ABC$
• D. cannot be determined due to insufficient information

Answer 1

The correct option is B is equal to the area of $△ABC$


$\text{Area}\left(△ABC\right)=\text{Area}\left(△PAB\right)+\text{Area}\left(△PBC\right)+\text{Area}\left(△PAC\right)$
$\Rightarrow \text{Area}\left(△ABC\right)=\frac{1}{2}\times cz+\frac{1}{2}\times ax+\frac{1}{2}\times by$
$\Rightarrow \frac{1}{2}(ax+by+cz)=\text{Area}\left(△ABC\right)$