How Can This Be True?

Figure ABC is made up of four pieces. Figure DEF is made of the same pieces, but in a different arrangement. In the DEF arrangement they appear to occupy an area that is 1 square unit less than in the ABC arrangement. How can this be?

Make up some puzzles like these and send them in with your solutions. We will post them on the SEED Web site.

Course:

Geometry [1]

Result/Solution(s)

Solution: How Can This Be True? Math Puzzle

Although ABC and DEF both appear to be triangles, they are not. On ABC, if you draw a straight line from A to B, the point M will be slightly below the line. On DEF, if you draw a straight line from D to E, the point P will be slightly above the line.

If AB were a straight line, then ANM would have to be similar to MOB; that is, the slope of AM would have to be the same as the slope of MB. But the slope of AM is

and the slope of MB is

If ABC were a triangle, its area would be

Similarly if DEF were a triangle, its area would be

But if we add up the areas of the pieces of ABC, we get a different result

red triangle

=

12

green triangle

=

5

L-shaped orange object

=

7

L-shaped green object

=

8

for a total area of 32 square units.

If we add up the areas of the pieces of DEF, we get 33 square units, the same as for ABC plus 1 more for the white square.

So, ABC is slightly less than a triangle, and DEF is slightly more than a triangle. Look closely.

Geometry puzzle [2]
Math Puzzle [3]

red triangle	=	12
green triangle	=	5
L-shaped orange object	=	7
L-shaped green object	=	8