Can anyone explain this to me? I’m stumped… 
I think it’s because the overall shape is not a perfect triangle, look at the green triangle, 5x2, ratio 2.5, the red one is 8x3, ratio 2.6667. if you hold a ruler against the slope of the top picture, you can see it bends inwards slightly, the bottom one bulges outward a little.
Am I right ?
Yeah, I’ve seen this before. The upper triangle is smaller, as the longest line is not straight on either one. The green and red triangles are slightly different sizes in the two pictures.
brian
I see what you guys are saying… but I still don’t see how that accounts for an entire empty square in the bottom picture.
[quote=“tigerman”]I see what you guys are saying… but I still don’t see how that accounts for an entire empty square in the bottom picture.[/quote]The bulge accounts for the extra square. Where did the bulge come from ? The emtpy square. Look at it a simpler way, replace the two triangles with rectangles of 5x2 and 8x3, and see how many empty squares there are, and how many squares used up, and you can see how it all adds up. They used triangles to fool you, it’s an optical illusion that the pieces make the same shape after moving them.
Or to put it another way, look carefully at where the diagonal line cuts through the squares of the graph paper. Especially where it goes close to the corner of a square you can see a different. EG follow the diagonal up form the bottom left. At the third square, there’s a little whit corner. See how much biggeer the corner is on the second picture. Now take that difference 13 times (for the 13 squares the line passes through). That can add up to 1 square.
Brian
I agree. The two ‘obviously equal’ triangles aren’t equal. If the diagonal of the red triangle in the upper diagram actually continued upward in a straight line, it would peak at 4.875 units, not 5.0 as shown. If the diagonal of the green triangle in the lower diagram continued upward in a straight line, it would peak at 5.2 units, not the 5.0 shown.
The imperceptible downward bulge of the top diagonal line of the upper ‘triangle’ eliminates the void of the lower ‘triangle.’
Because their endpoints are coincident, the diagrams appear equal but it’s only an illusion – much like something else we’ve all been discussing recently.