## shubhamsrg Group Title what is the ratio of area of a traingle ABC to the area of the triangle whose sides are equal to the medians of ABC ? one year ago one year ago

1. shubhamsrg

or,,prove that this ratio always equals 4/3

2. mukushla

im watching :)

3. estudier

|dw:1350313254678:dw|

4. estudier

x^2 + h^2 = c^2 and y^2 +h^2 = b^2, x+y = a x^2-y^2 = c^2-b^2 ......hmm might be better off with Cosine Rule:-(

5. estudier

Yukky algebra....think, think...

6. estudier

|dw:1350318976709:dw|

7. estudier

|dw:1350319309051:dw|

8. estudier

|dw:1350319508249:dw|

9. estudier

152 is a quarter 123 -> 123/235 is 4/3 -> 143/362 is 4/3 as well

10. estudier

Think that's right...

11. shubhamsrg

why is 352 half of 362 ?

12. estudier

Median divides triangle area in half?

13. shubhamsrg

how is 35 a median ?

14. ganeshie8

|dw:1350323581495:dw|

15. ganeshie8

72 || 34 coz 72 is the midsegment

16. ganeshie8

|dw:1350323706822:dw|

17. ganeshie8

7824 is a parallelog gram, now imagine we translate it down

18. ganeshie8

|dw:1350323784808:dw|

19. estudier

Yes, sorry, I didn't make the construction very clear.......

20. ganeshie8

it translates to 1672

21. shubhamsrg

why will 87 meet 6 ?

22. ganeshie8

diagonals bisect each other in ||gram

23. ganeshie8

so 65=52

24. ganeshie8

87 meets at 6, thats right, but before that we should prove this : 326 is the triangle made by medians

25. ganeshie8

87 meets at 6 becoz, 47 is congruent to 26, 42 is congruent to 21

26. ganeshie8

the only tricky part in this proof i see is proving this : 326 is the triangle made by medians

27. shubhamsrg

i still dont get 876 thing..

28. ganeshie8

ok did u get estudier diagram

29. shubhamsrg

yep..i followed every bit till now..

30. ganeshie8

he constructed, 26 \(\cong\) 47

31. shubhamsrg

yes..

32. ganeshie8

26 is just translation of 47, so they are || one more construction he did is, he translated other median also, the translated medians intersect at 6. (we can prove this, later)

33. shubhamsrg

ahh alright,,so 36 isnt a median yet..okay,,

34. ganeshie8

assume it is a median so that we prove the thing at hand, :)

35. shubhamsrg

alright,,

36. ganeshie8

72 || 84, agree ?

37. shubhamsrg

yes..

38. ganeshie8

becoz 72 is the midsegment

39. shubhamsrg

yes,,i could figure out..

40. ganeshie8

7284 is a ||gram i have constructed

41. shubhamsrg

7248*

42. shubhamsrg

yes,,clear till now..

43. ganeshie8

also, 74 congruent to 62, by construction

44. shubhamsrg

okay,,7426 is a //gm..

45. ganeshie8

now imagine we slide the ||gram 7248 along the side 41

46. shubhamsrg

so you're saying those 2 //gms are congruent..

47. ganeshie8

the diagonal in the ||gram 7248 translated to where ?

48. shubhamsrg

coincides with 62

49. ganeshie8

|dw:1350324847401:dw|

50. ganeshie8

the diagonal 74 coincides wid 62 the side 42 coincides wid 21

51. shubhamsrg

yes,,the 2 //gms are congruent..

52. ganeshie8

|dw:1350324939445:dw|

53. ganeshie8

7248 congruent 6127

54. ganeshie8

we just use this : 6127 is a ||gram that gives 5 is midpoint of diagonal 62

55. shubhamsrg

that one we already knew right ?

56. ganeshie8

which one ? i thought we're trying to prove 5 is midpoint of 62 lol

57. ganeshie8

whats ur question

58. shubhamsrg

we knew 5 was the mid point quite earlier..after that assumption, everything fitted..those 2 became congruent //gms and everything else..

59. ganeshie8

no

60. shubhamsrg

??

61. ganeshie8

those 2 became congruent || grams by construction, 5 got nothing to do wid it

62. ganeshie8

let me put everything in one reply

63. shubhamsrg

we assumed 36 to be the median..that gave 16 // 27 ..by which it was a //gm,, and thus 5 was the midpoint ( 2 diagnols intersected)

64. ganeshie8

ok if that make sense good :)

65. ganeshie8

so only assumption we need to prove is 36 is a median, right ?

66. shubhamsrg

yep..

67. sauravshakya

68. shubhamsrg

excellency yet again..!! thank you @sauravshakya

69. shubhamsrg

and big thank you to @ganeshie8 ..your idea was used here after all..

70. shubhamsrg

+ @estudier you sparked the figure, ofcourse ! :P

71. estudier

I managed to get an outline of this just with algebra --- what a mess!!