Question

Which of the following is true, if $AB=a,BC=b,CD=c$ and $DA=d$ ?

• A. $a+b=c+d$
• B. $a+d=b+c$
• C. $a+c=b+d$
• D. None of these

Answer 1

The correct option is C $a+c=b+d$

A circle with centre $O$, touches the sides $AB,BC,CD,DA,$ at $P,Q,R,S$respectively.
Join $OP,OQ,OR,OS$
Now two tangents drawn from an external C
point to a circle subtend equal angles at
the centre.
$\therefore \left(i\right)AP=AS,\left(ii\right)BP=BQ,\left(iii\right)CR=CQ,\left(iv\right)DR=DS$
$AP=AS$ (tangents from $A$) ....(*)
$BP=BQ$ (tangents from $B$) ....(*)
$CR=CQ$ (tangents from $C$) ....(*)
$DR=DS$ (tangents from $D$) ....(*)
Adding the relations given in (*), we get
$(AP+BP)+(CR+DR)=AS+BQ+CQ+DS$
$\Rightarrow AB+CD=(AS+DS)+(BQ+CQ)$
$\Rightarrow AB+CD=AD+BC$
$\Rightarrow a+c=d+b$
Hence choice (C) is correct answer.