The puzzle of extra square
Q
Observe the two figures ABC and PQR below.
The region inside them is coloured in red, blue, green and yellow. Their corresponding shapes and areas are equal.
The location of the green shape is same in the two.
But the other three shapes (coloured red, blue and yellow) are relocated in the figure at the right.
In this configuration, you will see that an extra square in white appears.
Can you figure out the reason ?
Ans:
Consider the side of each small square in the figures as one unit.
The two sides containing the right-angle, in the two figures, are of same length.
i.e., the bases are of 13 units while the heights are of 5 units.
Then where did the difference come from ?
Let us mark a point "Y" on both figures at the same location as shown.
If you notice keenly, Y in the figure at right is at a height of 2 units from the base.
The height of same Y in the figure at the left is less than 2 units !
This implies AC and PR are not of the same length.
But in two triangles, if two corresponding sides are equal, the triangles are congruent.
That is where the paradox is.
The two figures are not triangles at all !!!
AC and PR are in fact bent.
They appear to be straight to the human eye because of optical illusion.
So inferences should not be drawn using figures.
Only textual descriptions and the axioms of geometry should be used in reasoning.