Saturday, May 2, 2015

[Pri20150408YYS] Yin-Yang Semicircles?

Question


Introduction
     The diagram looks a bit like the Yin and Yang symbol, doesn’t it?  This problem can be solved easily using the correct insights.  I present two solutions: the first one is by direct calculation (in terms of p), and the second solution uses the powerful concept of ratio of similar figures.
     Whichever method is used, first we must make observations.  Can you see that there are three types of semicircles (small, medium and large)?  [H02, H04]

Let  S = area of small semicircle,  M = area of medium-sized semicircle,  and  L = area of large semicircle.

Note that  area of A  = area of C  = LM + S,  and  area of B = 2´(MS).  [H10, H11]

Solution 1 (by direct calculation)
     L = ½p(3)2 = 9p/2.   M = ½p(2)2 = 4p/2.   S = ½p(1)2 = p/2.
     area of A  = area of C  = 9p/24p/2 + p/2  =  3p
     area of B = 2´(4p/2p/2) =  3p
\ area of A : area of B : area of C  =  1 : 1 : 1.  (The areas are all the same)
Many pupils feel more comfortable using concrete approximations like  p  » 22/7  or  p  » 3.14,  but this tends to obscure relationships between entities, and makes the calculations messier.

Solution 2 (using similar shapes)
     An powerful idea is that the ratio of areas of similar shapes is the square of the ratios of their lengths.  When a figure is enlarged by a factor of (say) 5, we get a similar figure and the area becomes  52 = 25 times as large.  So  M = (2)2S = 4S  because the radius of the medium-sized semicircle is twice that of the small semicircle.  Likewise,  L = (3)2S = 9S.
     area of A  = area of C  = 9S – 4S + S  =  6S
     area of B = 2´(4SS) =  6S
\ area of A : area of B : area of C  =  1 : 1 : 1.  (The areas are all the same)

Commentary
     The second solution is neater because we do not need to deal with fractions or with pThe  ratio of areas of similar shapes is the square of the ratios of their lengths.  This concept is in fact required knowledge in secondary school mathematics, including GCE ‘O’ level “Elementary” Mathematics.

H02. Use a diagram / model
H04. Look for pattern(s)
H10. Simplify the problem
H11. Solve part of the problem


Suitable Levels
GCE ‘O’ Level “Elementary” Mathematics (“similar figures”)
* Primary School Mathematics (“areas”)
* other syllabuses that involve areas, ratios and or similar shapes.





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