Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

Points $P$, $Q$, $R$ and $S$ are chosen on the sides of parallelogram $ABCD$, so that $P$ is on line $AB$, $Q$ is on line $BC$, $R$ is on line $CD$, $S$ is on line $DA$, and $AP=BQ=CR=DS=\frac{1}{3}(AB)$. Compute the ratio of the area of $PQRS$ to the area of $ABCD$.

share|cite|improve this question
Um, is this an ongoing competition? It would be nice to be assured that answers here will be to late to be of any help with cheating. – Henning Makholm Oct 27 '11 at 0:54
Yes this competition has already been completed. No one in my class knew a fast shortcut to answer this problem, so that's why I came here. ;) – Tayyeb Din Oct 27 '11 at 0:56
I meant to comment, but forgot for a while. Sorry! There is something wrong with the problem. I can change the parallelogram and get different ratios. I am sure that with a little work you can also. My guess is that there was a transcription error, and we should have $BQ=(1/3)BC$, and similar things for the others. Then it is straightforward to give a general solution. – André Nicolas Oct 28 '11 at 19:45
up vote 1 down vote accepted

A nice trick for solving problems like this is that the desired ratio must be the same for any parallelogram with that construction, so we can choose a convenient one and just solve the special case.

enter image description here

Let ABCD be a square of side length 3; then PQRS is a square inside of ABCD excluding four right triangles in the corners, each of which has legs length 1 and 2. The area of ABCD is 9, and the area of the excluded triangles is $4(\frac{1}{2}*1*2) = 4$, so the area of PQRS is 5. Therefore the ratio of the area of PQRS to the area of ABCD is $\frac{5}{9}$.

share|cite|improve this answer
Thank you! But the question asked for the ratio of PQRS to the area of ABCD, not the interestion of PQRS and ABCD to ABCD. The answer was 5:9. – Tayyeb Din Oct 27 '11 at 0:43
PQRS is wholly contained in ABCD, so "the intersection of PQRS and ABCD" is the same thing as "PQRS". – Henning Makholm Oct 27 '11 at 0:59

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.