It can be seen that the hypotenuse of the large triangle is not a straight line, here is how you can calculate that:
Right triangles made within larger right triangles [like the green triangle in the overall triangle] have equal angles. So, the interior angles in all the triangles found in the diagram must be equal to each corresponding interior angle [green triangle's angle at the top of both green and overall triangle must equal the red's top most angle]. With other geometric knowledge, red triangle should be similar to green triangle.
We are not given angle measurements, so trigonometry is needed.
We do know the lengths of the legs of each triangle. So the basic trigonometric function of tan A = a/b, where both a and b represent the triangle's legs. We don't know the value of A (the angle) so we will need to use tangent's inverse, arctan [sometimes expressed as 1/tan x or tan x ^-1]
For shorthand, I will use the symbol "#" for an angle.
# = arctan (a/b)
For red triangle and green triangle to be similar, using # = arctan (a/b) must equal each other when used on both triangles.
For green triangle I will use # = arctan (a/b)
For red triangle I will use # = arctan (a'/b')
arctan (a/b) must equal arctan (a'/b') for these triangles to be similar, thus making the overall triangle really a triangle and not a quadrilateral.
Is this equation valid?
Does arctan (a/b) = arctan (a'/b')
arctan (2/5) = arctan (3/8)
after some calculations..
arctan (2/5) does not equal arctan (3/8)
This is means the angles are not equal to each other, meaning the triangles are not similar, meaning the whole triangle is quadrilateral. Because this picture is a quadrilateral and not a triangle, smaller triangles cannot be cut and pasted anywhere to form a solid triangle.
After all this, I feel as if I've used trigonometry wrong and that I am wrong. I haven't taken trig or done math in a while.
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