## imehs18 2 years ago Find the number of sides of each of the two polygons if the total numbers of sides of the polygon is 15, and the sum of the number of diagonals is 36.

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1. Decart

the number of sides is 30

2. imehs18

i was able to get the number of sides but i don't know how to show the solution

3. imehs18

the number of side is 9 and 6 i just need a solution how to get this

4. Decart

then the numbers of sides of the polygon should be polygons

5. kc_kennylau

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6. kc_kennylau
7. kc_kennylau

So the number of diagonals of a polygon with $$n$$ sides is $$\dfrac{n(n-3)}2$$

8. kc_kennylau

Why don't you set up equations :)

9. imehs18

number of diagonals i already given. i need a solution.

10. imehs18

can you help me?

11. kc_kennylau

You could try letting the numbers of sides of the two polygons be $$m$$ and $$n$$ respectively :)

12. imehs18

can you pls show me the solution. i got this m(m-3)/2 + n(n-3)/2 = 36

13. kc_kennylau

Ok I'll set up the equations for you but you'll work out the steps yourself :) $m+n=15$$\frac{m(m+3)}2+\frac{n(n+3)}2=36$

14. kc_kennylau

ok here's a hint: $$m=15-n$$

15. imehs18

m^2-3m+n^2-3n = 72 m^2+n^2-3(m+n) = 72 we know that m+n = 15 m^2+n^2-3(15) = 72 m^2+n^2 = 27 after that i dont know

16. Decart

substitute 15-n for m

17. imehs18

huh?

18. kc_kennylau

substitute $$m=15-n$$ to $$\dfrac{m(m-3)}2+\dfrac{n(n-3)}2=36$$ :)

19. kc_kennylau

into*

20. Decart

$\left( 15-n ^{} \right)^{2}-3\left( 15-n \right)+n ^{2}-3n=72$

21. kc_kennylau

yep, and this will be a quadratic :)

22. imehs18

im stuck

23. imehs18

i got -108 = -27n+n^2

24. kc_kennylau

25. kc_kennylau

26. imehs18

can you just pls provide me the whole solution im stuck. sorry if im stupid about this..

27. imehs18

i stop studying for 5years and i need to recall everything about geometry

28. Decart

2n^2-30n+108=0

29. Decart

(_n+_)(_n-_) factor

30. Decart

you did it right you just got 27 instead of 30

31. imehs18

how did you get the 30?

32. imehs18

got it

33. Decart

from the first equation 225-30n+n^2-45+3n+n^2-3n=72

34. Decart

combine like terms

35. imehs18

after that. what is next?

36. Decart

(15-n)^2 gave me the thirty

37. imehs18

how do i get the n now?

38. Decart

then you factor the 2n^2-30n+108 you can divide both sides by two to make it easier n^2-15n+54

39. Decart

two factors of 54 that total to -15

40. Decart

(x-_)*(x-_)=0

41. imehs18

(n-9)(n-6)

42. Decart

yes so those are your sides

43. imehs18

thank you so much for your patience with me. i really appreciate it.