Points $P$, $Q$, $R$, and $S$ are taken on sides $AB$, $BC$, $CD$, and $DA$ of square $ABCD$ respectively, so that \(\frac{AP}{PB} = \frac{BQ}{QC} = \frac{CR}{RD} = \frac{DS}{SA} =…

Question

Points $P$, $Q$, $R$, and $S$ are taken on sides $AB$, $BC$, $CD$, and $DA$ of square $ABCD$ respectively, so that $\frac{AP}{PB} = \frac{BQ}{QC} = \frac{CR}{RD} = \frac{DS}{SA} = \frac{1}{n}$. Then the ratio of the area of $PQRS$ to the area of $ABCD$ is

Accepted Answer

C - $1 + n^2 : (1 + n)^2$. With side s, each corner triangle has legs s/(n+1) and ns/(n+1), area n s^2/(2(n+1)^2). PQRS = s^2 - 2n s^2/(n+1)^2 = s^2 (1+n^2)/(1+n)^2.

Answer

A - $1 : (1 + n)$

Answer

B - $1 : n$

Answer

D - $(1 + n) : (1 + n^2)$

Points \(P\), \(Q\), \(R\), and \(S\) are taken on sides \(AB\), \(BC\), \(CD\), and \(DA\) of square \(ABCD\) respectively, so that \(\frac{AP}{PB} = \frac{BQ}{QC} = \frac{CR}{RD} = \frac{DS}{SA} =…

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