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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 ?2012-06-10
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    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).2012-06-10
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    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 ?2012-06-10
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    The idea is to cut the cube with a plane (say m) that is parallel to the faces BEGF and ADHI, containing the point C. You should note that the desired altitude of your triangle will lie on this plane m. The Pythagorean theorem is not being used on your original triangle, but rather the triangle lying on m where the hypotenuse is the desired altitude. The other two sides are the intersection of m with ABJI and HIFG respectively.2012-06-10
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    @aru I am a bit confused regarding your statement. From what i understand is that the plane m only contains the point C whereas the base of the triangle would still be on a plane perpendicular to plane m which is ABIF at edge AB. So how did u get a triangle on plane m ?2012-06-10
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    I added a picture to make the construction of the plane more clear.2012-06-10
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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.2012-06-10
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    I'm learning how to use Geogebra anyway; it was my pleasure.2012-06-11