can anyone explain what's happening here ?
- Thread starter Cheech
- Start date
Ah I got it. Its an optical illusion, or at least just close enough that no one notices it is off.
We are assuming that the hypotenuse of the overall large triangle is a straight line - it is not.
For this to work, the smaller triangles would have to be similar, i.e. all three angles would need to be equal, regardless of leg length. If you call the smaller acute angle of the grey triangle alpha, and the smaller acute angle of the red triangle beta, then tan(alpha)=tan(beta). If you calculate this out (tan = opposite leg length over adjacent leg length), tan(alpha) = 2/5 and tan(beta) = 3/8. 2/5 != 3/8. Basically, the gray triangle has a slightly steeper slope than the red triangle, but between the thickness of the black outline and the closeness of the angle, it looks like a straight line. The additional area taken up by that angle break in the hypotenuse is what gives the extra .5 units of area over what should be taken up by the larger outer triangle.
We are assuming that the hypotenuse of the overall large triangle is a straight line - it is not.
For this to work, the smaller triangles would have to be similar, i.e. all three angles would need to be equal, regardless of leg length. If you call the smaller acute angle of the grey triangle alpha, and the smaller acute angle of the red triangle beta, then tan(alpha)=tan(beta). If you calculate this out (tan = opposite leg length over adjacent leg length), tan(alpha) = 2/5 and tan(beta) = 3/8. 2/5 != 3/8. Basically, the gray triangle has a slightly steeper slope than the red triangle, but between the thickness of the black outline and the closeness of the angle, it looks like a straight line. The additional area taken up by that angle break in the hypotenuse is what gives the extra .5 units of area over what should be taken up by the larger outer triangle.
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Oh, it's only because I typed faster since I've been dying to use the word hypotenuse ever since I heard it on Family Guy.
Now to figure out how I'm going to fit interrobang into a conversation.
Wow.... so basically that extra little curveture of both triangles together with the thickness of the highlighted black line around the triangle actually makes up the extra square missing in that area.
That's messed up.
Thank goodness for smart people.!
That's messed up.
Thank goodness for smart people.!
if the hypotenuse is not a straight line then the object is not a triangle and that 'line' cannot be referred to as a hypotenuse.
Albeit, that is not the reason it doesnt work. it is a triangle but it is misrepresented in its size in your drawing.
to get more technical i made one of those to scale, one of your blocks = one inch in my drawing. note the line i drew across the top right point and the bottom left point. The triangles are not proportionate in size.
enjoy.
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and the surface area of the "extra" areas (beyond the curves) are equal to that of the empty box?
after a few minutes of messing around i came up with this^^^^^^^^
response to your last question....sure....why not
but keep in mind that all of those are straight lines so it would be the area of the obtuse triangle.
which should measure: Leg 1 8.5442 Leg 2 5.3851 Hypotnuse 13.9284 in.
figure out the area of that and you should have your answer
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