[UPDATE] Please review my progress, thoughts so far on 4-square problem

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LilWiz

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May 24, 2013
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Hi everyone :)

MYK1Qes.jpg


I'm stuck on (ii).. I can't figure it out. I breezed through the rest of the questions but I haven't got this yet. It needs to be done by Monday so I will work on it during that time and if I don't get it I will check back here. If anyone wants to help, can you try and give me just a hint rather than the full explanation? I would love to work it out "myself". Haha

I think it might have something to do with constructing other triangles? Maybe joining the two triangles on JF? Just confirming, I have done (i), but not (ii).
Thanks :)
 
Okay so I don't know if anyone is actually reading this but I know I can now solve the question if I prove that angle IHJ = angle HJI + angle CDI. I am struggling to prove this but I feel like I have made progress.
 
Why are people replying to other threads and not this one? Even if you don't know the solution I would like some feedback just so I know someone is out there!
 
LilWiz said:
Why are people replying to other threads and not this one? Even if you don't know the solution I would like some feedback just so I know someone is out there!
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You posted late in the evening; you posted multiple replies to yourself; you concluded in the "wee hours". Many ("most"?) volunteers were sleeping at the time and, by the time they (such as me) learned of this thread, it appeared already to have been dealt with. I'm afraid it's likely that many aren't even aware that you're the only previous poster on this thread. :(

At the moment, it's a Saturday morning and my child needs me to drive him around to various events. I'll return to this thread later, but it'll probably be a few hours. Please be patient! ;)
 
Okay well I'm from Australia so time zones are a bit different :) in any case I figured it out.

In the picture where I cut the triangles out and put them together, it is a right angled triangle. You can say the large triangle has internal angles IHJ, GHF and the other angle is the sum of the four angles HJI, CDI, CBG, and GFH. Now, you could say angle IHJ has an angle x and so GHF has an angle of 90 - x (because it's a straight angle in the original diagram, subtracting a right angle from the square, therefore the two angles add to 90). Now, since those are two angles of the large triangle, we can find the third (y) by saying
y = 180 - x - (90 - x)
y = 180 - x - 90 + x
y = 90
Therefore, the large triangle created from the four small triangles has a right angle at the vertex that all triangles meet at. This triangle has a hypotenuse 4d and a median b (splits the hypotenuse into two lengths of 2d). Now, there is a theorem that says that the median of a right angled triangle that meets the hypotenuse is half the length of the hypotenuse (basically because if you split the triangle along the median it creates two isosceles triangles, here is a great proof http://www.algebra.com/algebra/hom...right-triangle-drawn-to-its-hypotenuse.lesson). This means that b = (4d)/2, and b = 2d. So there you go
 
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