Question

If $D$ is any point on the side $BC$ of a $△ABC$, then

• A. $AB+BC+CA>2AD$
• B. $AB+BC+CA<2AD$
• C. $AB+BC+CA>3AD$
• D. None

Answer 1

Answer: (A)

$AB+BC+CA>2AD$

Given D is any point on BC of $△ABC$,
In $△ABD$,
$AD<AB+BD$ (I) (In a triangle Sum of two sides is always greater than the third side)
In $△ACD$,
$AD<AC+CD$ (II) (In a triangle Sum of two sides is always greater than the third side)
Adding I and II,
$2AD<AB+AC+BD+CD$
$2AD<AB+BC+CA$