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Question-209314




Question Number 209314 by SonGoku last updated on 06/Jul/24
Commented by SonGoku last updated on 06/Jul/24
How to find the of this polygon?
Howtofindtheofthispolygon?
Commented by mr W last updated on 07/Jul/24
you can only find the perimeter of  the polygon, nothing else of it.
youcanonlyfindtheperimeterofthepolygon,nothingelseofit.
Commented by SonGoku last updated on 07/Jul/24
So the only way to determine the diagonal of ang  irreular polygon, like the one in the image, is onlyo  thrugh practice? In other words, in the field?
Sotheonlywaytodeterminethediagonalofangirreularpolygon,liketheoneintheimage,isonlyothrughpractice?Inotherwords,inthefield?
Commented by Frix last updated on 07/Jul/24
Diagonal=d    18<d<35    ((√(197319))/4)<area≤((√(1278519))/4)       (min at d=35, max at d=((√(638290))/(29)))
Diagonal=d18<d<351973194<area12785194(minatd=35,maxatd=63829029)
Commented by Frix last updated on 07/Jul/24
For 1 triangle you need at least 3 measurements.  In this case, you need 1 additional measurment  for one of the triangles, the rest follows.
For1triangleyouneedatleast3measurements.Inthiscase,youneed1additionalmeasurmentforoneofthetriangles,therestfollows.
Commented by SonGoku last updated on 08/Jul/24
But, how did you at these calculations?
But,howdidyouatthesecalculations?
Commented by Frix last updated on 08/Jul/24
1^(st)  triangle 10, 28, d ⇒ 18<d<38  2^(nd)  triangle 15, 20, d ⇒ 5<d<35       ⇒ 18<d<35  Area of triangle a, b, c       =((√((a+b+c)(b+c−a)(a+c−b)(a+b−c)))/4)  Area of 1^(st)  triangle       A_1 =((√(−d^4 +1768d^2 −467856))/4)  Area of 2^(nd)  triangle       A_2 =((√(−d^4 +1250d^2 −30625))/4)  For the minimum obviously d=18 ⇒ A_1 =0  For the maximum I used differentiation
1sttriangle10,28,d18<d<382ndtriangle15,20,d5<d<3518<d<35Areaoftrianglea,b,c=(a+b+c)(b+ca)(a+cb)(a+bc)4Areaof1sttriangleA1=d4+1768d24678564Areaof2ndtriangleA2=d4+1250d2306254Fortheminimumobviouslyd=18A1=0ForthemaximumIuseddifferentiation
Commented by mr W last updated on 08/Jul/24
maximum area is when the  quadilateral is cyclic.  s=((a+b+c+d)/2)=((10+28+15+20)/2)=36.5  A_(max) =(√((s−a)(s−b)(s−c)(s−d)))     =(√((36.5−10)(36.5−28)(36.5−15)(36.5−20)))     =((√(1278519))/4)  see also Q30233
maximumareaiswhenthequadilateraliscyclic.s=a+b+c+d2=10+28+15+202=36.5Amax=(sa)(sb)(sc)(sd)=(36.510)(36.528)(36.515)(36.520)=12785194seealsoQ30233
Commented by SonGoku last updated on 09/Jul/24
Very sophisticated.   Congratulations  I willa study.  Thank you.
Verysophisticated.CongratulationsIwillastudy.Thankyou.
Commented by SonGoku last updated on 09/Jul/24
So, can I use this formula to determine the area of  any quadrilateral?
So,canIusethisformulatodeterminetheareaofanyquadrilateral?
Commented by mr W last updated on 09/Jul/24
no! it′s only for so−called cyclic  quadrilaterals.  generally a quadrilateral is not  uniquely defined when only its four  sides are given. you need an  additional condition.
no!itsonlyforsocalledcyclicquadrilaterals.generallyaquadrilateralisnotuniquelydefinedwhenonlyitsfoursidesaregiven.youneedanadditionalcondition.

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