Sometime back, I was chatting with this girl ‘SHE’. The name is not revealed as it may lead to identity revelation and unwanted interrogation and expectation and assumptions from fellow mates. I’ll tell you how this conversation started and ended in knowledge exchange and sharing which is the most desirable event in a conversation.
I was busy with the documentation work for my code testing. Suddenly a new mail alert popped out (Outlook Users understand this better) in my desktop. The Subject read – Solve this if u r genius. I opened the mail and found this one below
The mail asked me to find why there was a gap in the second triangle if the area of both the triangles, formed by rearranging the smaller pieces is same.
Since I was an engineering graduate and to the fact that, we had “Engineering Mathematics” for first two years with one maths paper for each semester, I looked into the figure more carefully, so as to dust my brains to check any traces of mathematics data existence which I had to study and work out for two years of my engineering life.
Whoa !! Some trigonometry question!! Hmmm….Not bad, my brain still works and it is active as before. !
I quickly remembered that the area of the triangle can be calculated by adding the area of the squares (Partially filled and fully filled. Filled with colors in this context). And found few squares which are not occupied by the triangle, so I counted the number of partial squares and found that they are same. I thought the area occupied by the triangle in the partial squares in both the figures may be different.
I pinged her…
ME: u know that ans?
SHE: No, u know?
ME: The area covered by the partially filled squares
along the hypotenuse is not same in both the figures.
So I think all those differences come into Single Square
when u re-arrange the pieces
SHE: mmm…do u think the hypotenuse line cutting the squares
are slightly different points in both the images?
SHE: But, between two points there can be only
one straight line.
(Oh Yes, That’s true, we cannot have more than one straight line in between two different points. Point noted !! So how can this be. I started thinking again)
ME: Yea, but u see the slopes are different in both the
cases as the angle of inclination of the hypotenuse to the
base is different.
SHE: But how? They both are right triangles,
ME: The blue and red triangle forms the
SHE: yea. . . .both of them have hypotenuse forming the same line
. . . parallel sides. . both base and height so. . it should the same angle
of elevation…then. . how can it have diff slopes?
ME: Haan par the angle of inclination differs as the
length of the hypotenuse differs…
SHE: but. . still. .as length of hypotenuse differs. .
but proportionally the other two sides too…rt?
ME: u have to construct a cardboard model and see
…u’ll see that the long hypotenuse is just bent a bit…..
if u r not sure….
SHE: ok. . but.. how is it theoretically possible. .
The base lines of the triangles in both figures are parallel,
the hypotenuse are parallel. .which means. . They should have
same angle of inclination / slope rt?
ME: mmm….it is optical illusion may b….
see this link
ME: thats wat it says…i told u na….the hypotenuse
lines are different in both cases…..wiki says in a
different way…it says it forms a quadrilateral
SHE: for the quadrilateral. . its understandable,
because. . The length of the sides gets increase,
so. .it was obvious dat there should be a gap somewhere
….for here, the base length and height are still the same and
if u look at the first diagram, which is sort of animated they flip
the blue and red triangles, the blue one exactly fits into the red one
’cause they are proportional. . dat is…of course only a visual thing. .
actual measurements can differ..
But. . I don’t see a reason y they shouldn’t have the same
slope / angle of slope
ME: hmmm…yea…but visually they appear proportional…
but wen u chk with a card board or chart paper model
u’ll know !! even I’m not satisfied with the wiki thing…
only wen u chk practically u will know..
SHE: OK.. I’m getting it. .. but . . .
y wont it have same slope????
ME: Equation of straight line, y-y1 = m(x – x1)
m = (y – y1)/x – x1)
‘m’ is slope…according to my knowledge, since the difference
between x and x1 & y and y1 differs, the slope differs..
SHE: :S…dunno….will have to actually calculate
ME: I did
for red, m = 3/8
for blue, m = 2/5
it differs very slightly…
0.375 and 0.4
Exactly….so the combined hypotenuse line doesn’t cut the
squares at same points in both the figures.
doubt cleared uh?
now i got a gud topic to blog !!
ME: I’ll put the conv in blog with some edit !!
ME: disguise the names
SHE: u r gonna blog about a missing square??
ME: u havent seen chat blogs??
SHE : ???
nope. . not much
this will be tirst one i guess
when u write and post
ME: hey u wud have seen that
lots of ppl rite
SHE: not really
I have seen blogs where ppl mention a few lines from a chat
and write the rest on it
or mention the line somewhere in the chat
my frnd told me. . .etc
anyways..see wen i post
SHE: hehe…all right
In the end of the conversation, I could conclude that the slope thing was there and the hypotenuse is not cutting the partially filled squares at same points in both the figures. So the remaining area of the partially filled squares along the hypotenuse of figure 1, comes as a single unfilled square in the figure two when the shapes are rearranged!
So with this, conversation ended for a few minutes. I really do not know if we have come to a conclusion about this missing square. Or this is still an unsolved one. But both of us were satisfied that we found an answer for the missing square and we were able to justify our conclusions with the mathematical proof.
Out conclusions may be wrong….If u have any other answers, please post it in the comments!!!