## eroshea 3 years ago Find the number of sides of each of the two polygons if the total number of sides of the polygons is 13, and the sum of the number of diagonals of the polygons is 25.

1. hartnn

polygon of n sides have how many diagonals ?

2. eroshea

[n(n-3)]/2 ... is that it?

3. hartnn

yes, so u could u form 2 equations using this formula?

4. eroshea

how?

5. hartnn

take side of one polygon as x and other as y x+y=13 [x(x-3)]/2+[y(y-3)]/2 = 25

6. eroshea

after that?

7. hartnn

can u work out second equation and tell me what u get as x^2+y^2= ?

8. eroshea

got it :)) the two polygon has 5 and 8 sides

9. hartnn

yes! that is correct. u had options ?

10. eroshea

i guess none... im just finding the sides of the two polygons is there a possible option for this problem? if there is, can you tell me what it may be?

11. hartnn

by option , i meant choices.

12. eroshea

owh.. hahaha! my bad.. none :)

13. hartnn

then u worked it out to get 5, 8 ? nice!

14. eroshea

yes! :)) you helped me a lot there..

15. pjaesecarilla

hello there :) can i ask @hartnn what you meant by "can u work out second equation and tell me what u get as x^2+y^2= ?" sorry i also encountered this problem on my solid mensuration

16. hartnn

so, did u get how i got this : [x(x-3)]/2+[y(y-3)]/2 = 25 ?

17. hartnn

when sides is 'x' number of diagonals is x (x-3)/2 same for y and total number of diagonals = 25

18. hartnn

that can be simplified as $$[x(x-3)]/2+[y(y-3)]/2 = 25 \\ x^2-3x+y^2-3y=50 \\ (x^2+y^2)-3(x+y)=50$$ but we know x+y =13 so, $$x^2+y^2-3\times 13=50$$ we can easily find x^2+y^2 from here.

19. hartnn

and once we know x^2+y^2 and x+y , we can find x and y you know how to ? or need help with that ?

20. pjaesecarilla

21. hartnn

ok, so we use the fact that (x+y)^2 = x^2+y^2 +2xy from this we will get xy. right ? then (x-y)^2 = x^2+y^2 -2xy from this we will get x-y , ok ? and now we have x-y, x+y if we just add then, 'y' gets eliminated and you get the value of x try once, if you get stuck, i will show you the entire solution

22. pjaesecarilla

ummm is xy=40?

23. hartnn

yes ! it is :) xy =40 is correct. what u got for x-y ?

24. pjaesecarilla

i kinda got stuck on that one. (x-y)^2 = x^2+y^2-2xy x^2=89-y^2 so. 89-y^2 + y^2 - 2(40) so i got 9. i dunno if im correct :/

25. hartnn

see, (x-y)^2 = x^2+y^2 -2xy = 89 -2(40) = 9 so, x-y = 3

26. hartnn

x+y = 13 x-y =3 just add them

27. pjaesecarilla

OOOOOOOOOOOOHHHHHHHHH!!!!!!! ok thanks man VERY MUCH haha i get it now :))))

28. imehs18

29. imehs18

30. hartnn

x+y = 13 x^2+y^2 -39 = 50 , so x^2+y^2 = 50+39 = 89 (x+y)^2 = x^2+2xy +y^2 13^2 = 89 + 2xy 169 = 89 +2xy 2xy =169-89 = 80 xy =40 :)

31. stephengmt

where did the xy come from? :) pls help thaaaaaaaaanks