Why would you ever post a math puzzle like this without linking to the solution?

It took me a little while, but here’s my solution:

The area of the total triangle is always 32.5 (5*13*0.5).

The area of the red triangle is 12 (3*8*0.5).
The area of the teal triangle is 5 (2*5*0.5).
The area of the yellow shape is 7.
The area of the green shape is 8.

Now we have a problem. 12+5+7+8 is 32. How is it that the overall area becomes 33 when the shapes are rearranged?

The problem is that the puzzle is bogus.

The area changes because the triangles are not congruent. Their angles are slightly off which causes a unit of slip in either direction spread out across the hypotenuse of both triangles.

The proof is in the angles of the smaller triangles versus the angles of the larger ideal triangle. Solving the triangle for it’s angles reveals the problem.

The Red triangle’s angles are 20.56° and 69.44°
The Teal triangle’s angles are 21.8° and 68.2°
All the triangle’s angles should be 21.04° and 68.96°
(all numbers rounded for clarity)

Another clue is that the leftover rectangular area defined by the two triangles is originally 3*5 units for an area of 15. When the triangles are rearranged, the area becomes 2*8 units, an area of 16.

If the ‘puzzle’ was legit, the red triangle would be 8*3.077 with an area of 12.31 and the teal triangle would be 5*1.92 with an area of 4.81. The magic and certainty of numbers prevail, I hate these kinds of carnival tricks.

My sophomore year of high school I received a “D” grade in Pre-Algebra. I was bored.

These automated triangle solving tools made this go much faster. Here is an MySQL query which returns the angles of the bogus triangles:

SELECT 
ATAN(5/13) * (180/PI()) AS whole_triangle_A,
90 - ATAN(5/13) * (180/PI()) AS whole_triangle_B, 
ATAN(3/8) * (180/PI()) as red_triangle_A, 
90 -ATAN(3/8) * (180/PI()) as red_triangle_B, 
ATAN(2/5) * (180/PI()) AS teal_triangle_A,
90-ATAN(2/5) * (180/PI()) AS teal_triangle_B;

(via Making Light)