How Can This Be True - Answer
1. For both of the pictures to be exactly the same, they must both
have the same area AND each of the pictures must have same area as the
area of the "PERFECT" triangle (oh no - shades of dreaded Math come to mind!) ...
Here are the calculated areas 4U:
Hint: the area of a triangle = 1/2bh
where b is the base and h is the height
1A. Before looking at either picture, calculate what the area of this picture
would be if it was a PERFECT triangle (i.e. 13 X 5 squares)
Area = 1/2bh = 1/2(13)(5) = 65/2 = 32 1/2
1B. Area of the top picture
Area = 1 + 2 + 3 + 4
= 1/2(5)(2) + 1/2(8)(3) + 7 + 8
= 5 + 12 + 7 + 8 = 32
1C. Area of the lower picture
Area = 1 + 2 + 3 + 4 + 5
= 1/2(5)(2) + 1/2(8)(3) + 7 + 8 + 1
= 5 + 12 + 7 + 8 + 1 = 33
Neither of the two pictures form a perfect triangle,
but both of them look very close to the outline of a perfect triangle!
2. Compare the slopes of the triangles and you realize that
the hypotenuse of the overall triangle is not a straight line ...
Hint: the slope of a line = rise / run
(the slope of a horizontal line is zero)
2A. Slope of the PERFECT triangle
Slope = rise / run = 5 / 13 = 0.384615
2B. Slope of Triangle 1 (in both pictures)
Slope = rise / run = 2 / 5 = 0.4
2C. Slope of Triangle 2 (in both pictures)
Slope = rise / run = 3 / 8 = 0.375
Prove it by blowing up the images and running a ruler along the
hypotenuse of both triangles - both are not a perfect line (but close!)