Chang's Math blog: June 2012

Saturday, June 30, 2012

Sec1 Circles

Click here for my solution

Warning:
Expressing OM in the correct form is crucial!
OM can be expressed as 10-r or sqrt(r^2 - 16) ;
If expressed in the former, the problem can be solved easily.
If expressed in the latter, however, you will be stuck!

Wednesday, June 27, 2012

Can u explain? ü (hint: Look at the gradient of the 2 triangles in each of the figure)

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My own attempt here:

The crux of the trick is that both the top and bottom figures are NOT triangles that most people would think they are!

The slope of the hypotenuse of the red triangle is 3/8 (or 0.375), while that of the blue triangle is 2/5 (or 0.4). Knowing these facts would have been end of story for demystifying this trickery...

But wait! We have merely conjectured that the "non-straightness" of the "hypotenuses" must have contributed to the "missing square".

Well, here's the proof that the conjecture is true:

The "hypotenuses" of both the figures are actually made up of 2 lines of different slopes. In the top figure, the effect is like a trench, while in the bottom figure, the effect is like a bulge upwards.

I have exaggerated the slopes here:
https://docs.google.com/open?id=0ByroEWpsGCWEVGNtRDNXTjZDdGs

In fact, through the link you have just seen, I also require the viewer to appreciate that the area of 2 triangles are the same if the two triangles share the same base and have their apex on the same line parallel to the base. This will be the premise where I prove that the area encompassed by the "trench" and the "bulge" (which is incidentally a parallelogram) equals to the area of the "missing square".

The following is the graphical proving by construction:
https://docs.google.com/open?id=0ByroEWpsGCWEVU13ZXBPbFRCdnc

Hope you have enjoyed the proving. :)

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Another version reveiled by Mr "luapnagle".
by luapnagle on Sep 1, 2010:
http://www.youtube.com/watch?v=ExUV3GOTDqE