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 ?

- shubhamsrg

- chestercat

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- shubhamsrg

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

- anonymous

im watching :)

- anonymous

|dw:1350313254678:dw|

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## More answers

- anonymous

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:-(

- anonymous

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

- anonymous

|dw:1350318976709:dw|

- anonymous

|dw:1350319309051:dw|

- anonymous

|dw:1350319508249:dw|

- anonymous

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

- anonymous

Think that's right...

- shubhamsrg

why is 352 half of 362 ?

- anonymous

Median divides triangle area in half?

- shubhamsrg

how is 35 a median ?

- ganeshie8

|dw:1350323581495:dw|

- ganeshie8

72 || 34 coz 72 is the midsegment

- ganeshie8

|dw:1350323706822:dw|

- ganeshie8

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

- ganeshie8

|dw:1350323784808:dw|

- anonymous

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

- ganeshie8

it translates to 1672

- shubhamsrg

why will 87 meet 6 ?

- ganeshie8

diagonals bisect each other in ||gram

- ganeshie8

so 65=52

- ganeshie8

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

- ganeshie8

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

- ganeshie8

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

- shubhamsrg

i still dont get 876 thing..

- ganeshie8

ok did u get estudier diagram

- shubhamsrg

yep..i followed every bit till now..

- ganeshie8

he constructed,
26 \(\cong\) 47

- shubhamsrg

yes..

- 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)

- shubhamsrg

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

- ganeshie8

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

- shubhamsrg

alright,,

- ganeshie8

72 || 84, agree ?

- shubhamsrg

yes..

- ganeshie8

becoz 72 is the midsegment

- shubhamsrg

yes,,i could figure out..

- ganeshie8

7284 is a ||gram i have constructed

- shubhamsrg

7248*

- shubhamsrg

yes,,clear till now..

- ganeshie8

also, 74 congruent to 62, by construction

- shubhamsrg

okay,,7426 is a //gm..

- ganeshie8

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

- shubhamsrg

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

- ganeshie8

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

- shubhamsrg

coincides with 62

- ganeshie8

|dw:1350324847401:dw|

- ganeshie8

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

- shubhamsrg

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

- ganeshie8

|dw:1350324939445:dw|

- ganeshie8

7248 congruent 6127

- ganeshie8

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

- shubhamsrg

that one we already knew right ?

- ganeshie8

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

- ganeshie8

whats ur question

- shubhamsrg

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

- ganeshie8

no

- shubhamsrg

??

- ganeshie8

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

- ganeshie8

let me put everything in one reply

- 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)

- ganeshie8

ok if that make sense good :)

- ganeshie8

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

- shubhamsrg

yep..

- anonymous

##### 2 Attachments

- shubhamsrg

excellency yet again..!! thank you @sauravshakya

- shubhamsrg

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

- shubhamsrg

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

- anonymous

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

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