Changing   areas

Here is a little puzzler that will keep you thinking for a while.   On the surface, it looks impossible for the identical four pieces to form both a rectangle and a triangle, but with different areas.   However when you are finished with the We start out with a 5x5 grid matrix and draw lines through the figure as shown below.

     Here, on the left side, is the 5x5 starting grid guide. It is 5" high and 5" wide and it is marked off in 1 inch increments to allow an easy layout of the puzzle. 1" increments will work out just fine for this exercise. After you get your grid made up then you are ready to divide it up with 3 straight lines, into four pieces.

     Next step is to mark off the grid the way it is shown above.   When you are finished you will have two triangles that measure 2" at the base and are 5" high.  You also will have two trapezoids, with 2" tops and 3" bases which are also 3" in height.   These pieces can now be cut out and assembled into a large triangle which is shown in the illustration below.


These four pieces can be assembled into a triangle or a rectangle (it's original configuration) however, when viewed as a triangle, the area is one square inch less tan when viewed as a rectangle. Hmmnn? something funny going on here...

Here, on the left you can see the pieces which you have cut out of the 5" x 5" grid.   Please note that the dimensions indicate that if you put these four pieces together then you will make one large acute triangle that measures 6" at the base and is 8" in height.   The formula for calculating the area of a triangle is : Base times height, and divide that product by two.   Or written out : " .5 x BH ".   So lets do the math on the composite triangle. The base is 6" and the height is 8" so 6 x 8 = 48. If we divide 48 by 2 (the same thing as multiplying by .5) then we discover that our triangle has an area of 24 sq. in.   Half of 48 is 24.   But hang on a second, here... didn't we make these pieces from a piece of carboard that was 5" x 5" ?  The formula for the area of a rectangle is base times height. So if it is 5 x 5 then it would be 25 sq. in.   Duh, isn't that one inch bigger in area? It seems like we lost a whole square inch in the rearranging of the pieces.  Double check by making a square out of them once more to make sure it still measures 5" x 5" , then put them together as a triangle again and measure that once more.  Hmmnn? Somthing funny going on here.... Can you tell what it is ?

     Now for the question, where in the world did that missing square inch disappear to?


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