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estudier
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|dw:1350208678915:dw|
sauravshakya
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What is that?
Jonask
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is that 5,7,10
estudier
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A square, a point inside of it distances from 3 vertices as shown
estudier
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9,10,7
estudier
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This is for experimentX, who loves algebra..:-)
experimentX
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haha ... you are kidding!!
estudier
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It's not too hard..(it's still too early)
experimentX
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is that all info??
Jonask
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if we call ne side a
then the area is
\[A=a^2\]
Jonask
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and the side is less than 17
Jonask
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according to triangle equality
experimentX
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|dw:1350209046775:dw|
Jonask
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\[9^2=a^2+10^2-2(10)(a)\cos \theta\]
\[762=10^2+a^2-2(a)(10)\cos (90-\theta)\]
estudier
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2 var 1 equation suggests simultaneous equations as the way to go....
hartnn
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|dw:1350209083317:dw|
Jonask
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i meant 7^2|dw:1350176811646:dw|
estudier
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JonasK is on the right track
Jonask
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what about using the area rule
experimentX
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probably @hartnn method would work out too ... since they are all independent equations.
estudier
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Probably...
estudier
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"what about using the area rule"
U just want to get rid of the 90- bit and then you can get rid of theta
Jonask
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yes
experimentX
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still that's quite a bit of algebra since they are all non linear equations ... let's try something better.
Jonask
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okay
estudier
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Cos of angle opposite 7 is sin theta....
experimentX
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|dw:1350209708034:dw|
estudier
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What I mean is that if angle opposite 9 is theta, then cos of angle opposite 7 is sin theta
estudier
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As JonasK had it at first...
estudier
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expX, now eliminate theta....
Jonask
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\[32=-20(a)\cos \theta+20(a)\sin \theta\]
estudier
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Don't be lazy....:-)
experimentX
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haha ..
estudier
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Sin and Cos should auto response.....
Jonask
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\[\frac{ 32 }{ -20 a}=\cos \theta -\sin \theta\]
Jonask
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\[\cos \theta -\sin \theta =\sqrt{2} \sin(45-\theta)\]
Jonask
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with - before
estudier
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@Jonask An identity (well known) in Sin and Cos?
experimentX
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haa ... i was being fool trying to eliminate theta directly.
experimentX
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|dw:1350210145598:dw|
estudier
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A bit messy but sufficient...:-)
Jonask
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seems great
experimentX
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still quartic equation is pretty awful ... i must say!!
estudier
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Yes, squaring and adding gets shot of theta...
estudier
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U get after simplification...400x^2 = (x^2 +19)^2 + (x^2 +51)^2 ->
x^4 -130 x^2 +1481
estudier
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Which is quadratic in x^2
experimentX
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Oh .. that's pretty nice!!
experimentX
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direct answer.
Jonask
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let \[-10^2-a^2=b\]
\[\frac{ (7^2 -b)^2}{ (-20a )^2}+\frac{ (9^2-b)^2 }{ (20a)^2 }=1\]
Jonask
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@estudier seems better
shubhamsrg
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|dw:1350210434675:dw|
x+2y = 7 ..(1)
y^2 + (x+y)^2 = 81 ..(2)
hope that does it..
shubhamsrg
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correction :
x+2y = 10
shubhamsrg
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solving shouldnt be much problem since from (1) , x+y = 10-y
and we plug that directly in (2)
Jonask
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the line with lenght 10 is not necessarily a diagonal when extrapolated ??
Jonask
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not sure
estudier
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Eh? Some extra ssumptions there....
shubhamsrg
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ohh..lol..so sorry! :P
Jonask
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no prob
Jonask
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\[a=10,834, a=3,552\]
Jonask
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can we eliminate according to triangle inequality 3,552
such that \[A=(10,834)^2\]
estudier
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65 + 14 sqrt 14 I think.
Jonask
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same thing i just approaximated it
Jonask
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\[A=(\sqrt{64+14\sqrt{14}} )^2\]
Jonask
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\[A=65+14\sqrt{14}\]
Jonask
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@shubhamsrg if we consider the triangles to be equilateral then how wuld we solve it
|dw:1350178935156:dw|
with common side 10,
shubhamsrg
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equilateral ?
Jonask
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|dw:1350179036162:dw| such that the legth is now diagonal
shubhamsrg
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you mean like this:
|dw:1350211506937:dw|
??
Jonask
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yes
shubhamsrg
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for that i had given the solution above..
x+2y = 10 ..(1)
y^2 + (x+y)^2 = 81 ..(2)
from (1),
x+y = 10-y
and we plug this in (2) to solve for y, and x and ultimately area..
sauravshakya
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Jonask
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nice work @sauravshakya
sauravshakya
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Thanx @Jonask
Jonask
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what disqualifies 65-14root14
estudier
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@Jonask That's for x^2, remember....
sauravshakya
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I think I have written reason for that too.
estudier
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@sauravshakya Yes, good job (I think he just forgot it was for x^2)