Question

Fill in the blanks to make the statement true.

The sum of three altitudes of a triangle is __________ than its perimeter.

Answer 1

In $△ABC,$ $AD,BE,CF$ are the altitudes of sides $BC,AC,AB$ respectively.

We know that, in all the line segments that can be drawn to a given line from a point not lying on it, the perpendicular line segment is the shortest.

Now, $AD\perp BC⟹AB>AD$ and $AC>AD$

$⟹AB+AC>2AD$ ........... $\left(i\right)$

$BE\perp AC⟹AB>BE$ and $BC>BE$

$⟹AB+BC>2BE$ ........... $\left(ii\right)$

Similarly, $CF\perp AB⟹AC>CF$ and $BC>CF$

$⟹AC+BC>2CF$ .......... $\left(iii\right)$

Adding $\left(i\right),\left(ii\right)$ and $\left(iii\right)$ we get

$2(AB+BC+AC)>2(AD+BE+CF)$

$⟹AB+BC+AC>AD+BE+CF$

Hence, sum of three altitudes of a triangle is less than its perimeter.