If $\dfrac{(a + b)}{(b + c)} = \dfrac{(c + d)}{(d + a)}$, which of the following statements is always true?

Question and options

\dfrac{(a + b)}{(b + c)} = \dfrac{(c + d)}{(d + a)}

, which of the following statements is always true?

$a + b + c + d = 0$
$a=c,$ or $a+b+c+d = 0$
$a=c$
$a=c,$ and $b=d$

Answer and explanation

Correct answer: B - $a=c,$ or $a+b+c+d = 0$

Cross-multiplying: (a+b)(d+a)=(c+d)(b+c) simplifies to a^2-c^2+b(a-c)+d(a-c)=0, i.e. (a-c)(a+b+c+d)=0. So a=c or a+b+c+d=0; neither alone is always true.

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If \(\dfrac{(a + b)}{(b + c)} = \dfrac{(c + d)}{(d + a)}\), which of the following statements is always true?