Find the Fallacy

2004-10-19T04:03:00.000-07:00

How to prove a right angle congruent to an obtuse angle?

Please click the following link for the geometrical figure:
http://home.att.net/~numericana/answer/false.gif

Consider the above picture: ABCD is a rectangle and E is a point near D slightly outside the rectangle so that AE is equal to AD. H is the middle of CD and K is the middle of CE. The perpendicular to CD going through H and the perpendicular to CE going though K intersect at a certain point J.
Now consider the sides of the triangles BCJ and AEJ: First, BC=AE (since both of these are equal to AD). Second, JB=JA (J is on the perpendicular bisector of CD, which is also that of AB). Third, CJ=EJ (J is on the perpendicular bisector of CE, by construction). The inescapable conclusion (I swear it's true!) is that BCJ and AEJ are congruent triangles (as their 3 sides are congruent), therefore the angles CBJ and EAJ are equal...
The angles JBA and JAB are equal (since JAB is an isosceles triangle), the picture clearly tells you that if you subtract JBA from CBJ and and JAB from JAE you obtain ABC and BAE. Yet one of these (ABC) is a right angle, whereas the other (BAE) is obtuse by construction. How can this be? Find the flaw or the fallacy here?

Answer:

The geometrical figure is not possible to draw and hence misleading. Hence the argument depending upon the assumption of the possibility of the drawing leads us to the wrong conclusion. The possible right geometrical figure could be seen here:
http://home.att.net/~numericana/answer/solution/true.gif

The point J is so far out that the line JE is outside the angle AJD, not inside it, as the bad picture led you to believe...

Must be Red colour

2004-10-11T04:28:00.000-07:00

Three men are buried in the sand all facing forwards with their heads above ground. Each man has a hat placed on his head selected from a bag containing 3 red hats, and 2 black hats. The men cannot turn around to see the men behind them. The man at the back is asked what hat he is wearing. He replies 'I do not know'. The middle man is asked what hat he is wearing. He also replies 'I do not know'. The man at the front is then asked what hat he is wearing. He replies 'I am wearing a red hat'. How did he know? One can safely assume here all the three are good at logic.

Answer:

Since the man at the back could not determine his own hat, this means that the front two men could not have been wearing black hats and therefore, there must be at least one red hat on the two front men. So the middle man must not be able to see a black hat otherwise he would know he had a red one on. Therefore the front man must be wearing a red hat, which finally he deduces. Interestingly, the other two can never determine their own hats.

Must be Red Colour

Find the Fallacy

Must be Red colour