Find the shaded area | Poland Math Olympiad Geometry Problem

In this video, we have been given that ABCP and PQST are a square and area of APM is 9, area of PCBM is 27. Then we have to find shaded area. So, ABCP, it is a square. That means area of ABCP, it will be APS square and area of ABCP, it is 9 minus 27. It is 9 minus 27. It is APS square and 36, it is APS square. So, AP, it will be 6. AP is 6 and this angle, it will be 90 degrees and now area of APM, it is 9 and area of APM, it will be 1 over 2 times base is AP and height is AM. It is 9 and it is 1 over 2 times AP is 6 times AM. AM, it is 9 and 2 times 3 is 6 and 3 times 3 is 9. So, AM, it is 3. AM, it is 3 and ABCP, it is a square. That means AB, it will be equal to AP. That will be 6. So, BM, it will APM, PMS square, it will be APS square minus AMS square by Pythagoras theorem and PMS square, it is 6S square minus 3S square and PMS square, it is 36 minus 9 and PMS square, it is 9 times 5. So, PM, it will be root under 9 times 5. That will be 3 times root 5. PM, it is 3 times root 5 and now this angle, it will be 90 degrees and angle AMP, it will be equal to angle TMB. That will be opposite angles. These two angles, they will be equal and interangle AMP and interangle TMB, angle AMP, it is equal to angle TMB and angle MAP, it is equal to angle MTB. That is 90 degrees. So, interangle AMP, it will be similar to interangle TMB. Hence, AM over TM, it will be equal to MP over MB. It will be equal to 6 over TB and 3 will get cancelled. So, 3 over TM, it is root 5. It is 6 over TB and we have 3 over TM is root 5. So, TM, it will be 3 over root 5 and we have root 5 is 6 over root 5 and PQST, it is a square. That means, PQ, it will be equal to PT and PQ, it is PT, it is PM minus MT and PQ, it is PM is 3 times root 5 minus MT is 3 over root 5 and PQ, it will be 15 minus 3 over root 5. That will be 18 over root 5. PQ, it is 18 over root 5 and now, shaded area, it will be area of PQST minus area of PCBM minus area of MBT and area of PQST, it will be PQ, a square minus area of PCBM is 27 minus area of MBT, it will be 1 over 2 times base is MT and height is BT and PQ is 18 over root 5, a square minus 27 minus 1 over 2 times MT is 3 over root 5 times BT is 6 over root 5 and 2 times 3 is 6. So, it is 324 over 5 minus 27 minus 9 over 5 and it is 324 minus 135 minus 9 over 5 and it is 324 minus 144 over 5 and it will be 180 over 5, that will be 36, it is 36.