Question

State true or false:

Sum of three sides of a triangle is less than the sum of its three altitudes.

• A. True
• B. False

Answer 1

The correct option is B False

Given: $△ABC$, AD, BE and CF are perpendiculars from A, B and C respenctively on the corresponding sides.
Now, In $△ADB$,
$AD<AB$ (Hypotenuse is the longest side)
Similarly in $△ADC$,
$AD<AC$ (Hypotenuse is the longest side)
Hence, $2AD<AB+AC$ (1)
Similarly we can say, $2BE<BC+AB$ (2)
and $2CF<AC+BC$ (3)
or adding (1), (2), (3)
$2(AC+AB+BC)>2(AD+BE+CF)$
$AC+AB+BC>AD+BE+CF$
Thus, perimeter of the triangle is greater than the sum of altitudes