I am busy writing
testtaking strategies for CAT 2007, CAT2008 aspirants and I thought I should meanwhile
share a small strange experience that happened to me in 2006. This particular
piece was written in 2006 itself and the events described actually happened.
This event left a very strong impression on me because from that day onwards I
started solving most of the problems mentally and without using pen or paper.
My day had ended and I was relaxing in my office with two
colleagues when one of them popped this question:
ABCD is a trapezium with AB and CD as the longer and
shorter of the two parallel sides, respectively. A circle is drawn with AB as
diameter such that it touches the side CD and bisects the sides AD and BC. What
are the angles of the trapezium?
As I saw the problem, the first thing that was obvious to
me, as it would be obvious to any average person, that everything was
symmetrical i.e. the sides AD and BC were inclined at the same angle to AB,
ABCD was an isosceles trapezium, and that the circle touched CD at its
midpoint. So far so good. Now what?
My next inspiration came after a couple of pencil drawings.
It looked like the figure shown below:
As AB is the diameter of the circle angle AEB =
90Â°. E is the midpoint of AD. In triangle ABD, since
E is the midpoint and BE is perpendicular to AD, triangle ABD is an
isosceles triangle with AB = BD.
What next?
I could not go further than this at that moment. I left it
there, deciding to visit the problem again at home in the evening.
I sat after dinner to look at the problem again. The good
thing about the problem was the friendly feeling it was giving that the
solution was right at the corner of my eye, and that I only had to see it in
order to solve the problem. My hopes didnâ€™t seem to be wellfounded however.
After several attempts, I was nowhere near to solving the problem.
My next try yielded the following result:
Let CD touch the circle at P. Let perpendiculars dropped from P and D on AB meet AB at O
and Q respectively. Let DE = EA = x.
It was clear that I was lost. I was nowhere near to
finding the finding the solution. I switched off the lights and went to sleep
thinking about the problem.
What happened next was bizarre. In my dream, I could
clearly see the problem that I was trying to solve with the whole figure right
in front of my eyes. Not only that, I had taken a completely new approach to
solving the problem. In my mind, I found myself drawing the figure shown below:
I knew that ABD was an isosceles triangle with AB = BD.
Hence AB = BD = 2r. P and F were midpoint of DC and BC. Therefore, PF // BD in triangle DBC. Also, PF = BD/2 = r. Let O be the centre of the circle. Hence, in triangle PFO, PO = FO = PF
= r. > angle FOP = 60Â° > angle FOB = 30Â° [angle POA = 90Â°] In > triangle FOB, FO =
BO = r and angle FOB = 30Â° > angle OAE =
75Â°.
Hence, the angles of the trapezium are 75Â° and 105Â°.
I woke up with a start. I had solved the problem. But now
I had a new problem to solve:
HOW DID I
MANAGE TO SOLVE A PROBLEM IN MY DREAM?!!!!!!!!!!!
