Question

If $D$ is any point on the side $BC$ of $\Delta ABC$, then which of the inequality holds true?

• A. $AB+BC+CA>2AD$
• B. $AB+BC+CA<2AD$
• C. $AB+BC-CA>2AD$
• D. $AB+BC+AC=2AD$

Answer 1

The correct option is A $AB+BC+CA>2AD$

Sum of two sides of a triangle is greater than the third side.

In $\Delta ABD$, $AB+BD>AD\dots \dots \left(i\right)$
In $\Delta ACD$, $AC+DC>AD\dots \dots \left(ii\right)$
On adding (i) and (ii),
$AB+BD+DC+AC>AD+AD$
But $BD+DC=BC$.
$\Rightarrow AB+BC+AC>2AD$