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!)
3. The bottom triangle has ~181.2 degrees when by definition a triangle must
have only 180 degrees. I used the inverse tangent function and applied it to
both angles that were not right angles. I calculated both what the angles should
be based on the lengths of the whole triangle at the bottom and what they
actually were based on the smaller triangles that formed the angles. The top
right angle of the bottom triangle should be ~68.96 degrees based on
tan^-1(13/5) but when you input the lengths of the smaller angle into this
formula (tan^-1(8/3)) the angle comes out to be 69.44 degrees. Likewise with the
other angle it should be ~21.04 degrees and actually is ~21.8. Larger angles
mean more area into the shape.
Conclusion: the 2 objects are NOT
triangles
(even though they at first appear
so!)
and they have different areas...
Thanks to Paul Stafford, Eric Nelson, Roy Bishop
& freeclimber!
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