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Each edge of the following cube is 1 and C is a point on the edge.

enter image description here

What would the height of triangle be in this case , how would you measure it?

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Assuming you mean "height" as "altitude passing through point $C$". You might want to clarify if this is not correct:

The height of the triangle is just the distance from $C$ to the line $AB$. Cutting the cube perpendicular to $AB$ and containing point $C$, we find a right angled triangle with both legs of length $1$, and the height of the triangle as its hypotenuse. So the height of the triangle is $\sqrt{1^2+1^2}=\sqrt 2$

It's also good to realize that the height doesn't change depending on where $C$ is, so you can just arbitrarily move it to one vertex and take the height along the corresponding face.

Picture:

enter image description here

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Are you saying that the height remains the same whether the point c is on edge HG or edge DE ? –  Rajeshwar Jun 10 '12 at 15:40
    
No, I think that Robert meant that it doesn't matter where on $\overline{GH}$ the point $C$ is (since that is where $C$ was specified to be). –  robjohn Jun 10 '12 at 15:46
    
I cant understand what Robert meant when he stated "Cutting the cube perpendicular to AB and containing point C, we find a right angled triangle with both legs of length 1, and the height of the triangle as its hypotenuse." I am confused on how you could use the phythagoras formula in a case such as this when the point is not on the same level as the base ? –  Rajeshwar Jun 10 '12 at 15:52
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I added a picture to make the construction of the plane more clear. –  Robert Mastragostino Jun 10 '12 at 17:25
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I just saw the picture and literally in less than a second I understood what you meant. Thank you so much for putting in the effort. –  Rajeshwar Jun 10 '12 at 17:35
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