## Wednesday, May 27, 2015

### [OlymPri20150527AGSQ] Three Angles and Three Squares

Question

Introduction
This question, from a Facebook forum, is essentially the same problem as the one in a previous article.  However, here we are not allowed to use any advanced mathematics like arctangent (inverse tangent).  It must be a solution that a primary school pupil can come up with, or at least understand.

Solution 1
Refer to the article “Angles with nice tans add up nicely”.  Ðb + Ðc  = Ða = 45°.  So  Ða + Ðb + Ðc  = 2 Ða = 90°.  But using this method is “cheating”.

Solution 2a
I happened to be watching this YouTube video that solves this problem.  What a coincidence!  The presenter is Professor Zvezdelina Stankova from the University of California, Berkeley.  I liked the way she used the “Act it out” heuristic by using a pair of scissors to cut out the angles on paper, and then putting them together to see if  they actually add up to  90°.  The trouble with this approach is: How do you know that they add up exactly to 90°?  So Stankova proceeded to give an elementary but rigorous proof.  Her proof is elegant, but I think the demonstration of the fact that  ÐEHD  is a right angle can be tightened using the idea of rotation.

Solution 2b
I am going to follow Stankova’s naming of the vertices, except that our problem is the mirror image of the one that is shown in the video.  Ðc  is easily seen to be  45°.  This is because  DBAE  is an isosceles triangle with  ÐBAE = 90°Ðc = (180° – 90°) ¸ 2 = 45°.  We extend the grid to a  2  by  3  grid, and construct  HD  and  HE

Obviously  HD  = HE = CE,  because they are all hypothenuses (that is the correct plural form, not “hypotheni”.  I checked)  of right-angled triangles with some length of  1  unit and another length of  2  units including the right angle in between.  Another way to think about it is that they are all diagonals of a  2  by  1  rectangle of some orientation.

Notice that  ÐIDH = ÐFHE = ÐECA = Ðb,  for the same reason.  Observe also that the rectangle  HCDI  is rectangle  HFEK  rotated by  90°  clockwise.  Every point and every line of the original rectangle is rotated by the same amount.  In particular,  HE  is rotated by  90°  to get  HD.  Hence  ÐEHD = 90°,  and since  DHDE  is an isosceles triangle,  ÐHDE = 45° = Ðc.   Ða + Ðb + Ðc = ÐEDA + ÐIDH + ÐHDE = ÐJDC = 90°©

Solution 3
One of the viewers, Ian Agol, contributed a solution, which I think is the most elegant solution.  It is a proof without words.  If you just stare at the picture and think, you should “get it” without explanation.  The diagram below is an adaptation of his solution.

If you want an explanation:  Ðc = ÐJDK  = 45°, obviously.  The red grid is constructed at a  45°  angle to the black grid, but has a bigger length.  Nevertheless,  ÐEDQ = ÐECA = Ðb,  because  DQDE  is just an enlarged version of  DACE.  [They are similar triangles.  The enlargment factor is Ö2,  but we do not need this.]  By looking at  ÐJDC,  you will see that  Ða + Ðb + Ðc  = 90°.

Remarks
Unlike the usual examination-based school “mathematical” practices, real mathematics is not just about getting the answer.  It is about creativity, seeking understanding and seeking elegance.  We prefer short and sweet solutions to long and complicated solutions.  Sometimes when we finish solving a problem, another solution or a few other solutions pop-up.  Having different approaches to a mathematical problem allows us to understand the problem and the inter-relationships better.

H01. Act it out
H02. Use a diagram / model
H04. Look for pattern(s)
H05. Work backwards
H09. Restate the problem in another way
H10. Simplify the problem
H11. Solve part of the problem

Suitable Levels
* other syllabuses that involve angles

#### 1 comment:

1. Drawing trend lines is one of the few easy techniques that really WORK. Prices respect a trend line, or break through it resulting in a massive move. Drawing good trend lines is the MOST REWARDING skill.

The problem is, as you may have already experienced, too many false breakouts. You see trend lines everywhere, however not all trend lines should be considered. You have to distinguish between STRONG and WEAK trend lines.

One good guideline is that a strong trend line should have AT LEAST THREE touching points. Trend lines with more than four touching points are MONSTER trend lines and you should be always prepared for the massive breakout!

This sophisticated software automatically draws only the strongest trend lines and recognizes the most reliable chart patterns formed by trend lines...

http://www.forextrendy.com?kdhfhs93874

Chart patterns such as "Triangles, Flags and Wedges" are price formations that will provide you with consistent profits.

Before the age of computing power, the professionals used to analyze every single chart to search for chart patterns. This kind of analysis was very time consuming, but it was worth it. Now it's time to use powerful dedicated computers that will do the job for you:

http://www.forextrendy.com?kdhfhs93874