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A base of a pyramid is a square of side 10. The lengths of three consecutive side edges are 8, 6, 10. Calculate the lenght of the fourth side edge.

I have not the slightest idea how to touch this. The triangle sides aren't isosceles so the line $|OO'|$ where $O$ is the apex and $O'$ is the point of the base diagonals crossing is not perpendicular to the base. Therefore, I simply don't know how to move on with this.. I see no angles to use some trigonometry on, nor any friendly right triangles which I could use to calculate something. Could you please help?

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You don't need to solve for $h$ (the height of the pyramid) at all. Nor do you need to know the dimensions of the base. Let $P$ be the point on the base that's directly below the vertex of the pyramid. If you write $a$, $b$, $c$ and $d$ for the distances from $P$ to each of the four edges, then you get

(1) ... $a^2+b^2+h^2=8^2$

(2) ... $b^2+c^2+h^2=6^2$

(3) ... $c^2+d^2+h^2=10^2$

(4) ... $d^2+a^2+h^2=l^2$

where $l$ is the length of the fourth side.

Now add (1) and (3), and subtract (2) and (4). This will give you $l$ easily.

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Thank you but where did you get these four equations from? As they use four numbers, I don't suppose it's Pythagorean theorem. – Straightfw May 6 '12 at 12:02
@Straightfw: It is the Pythagoras theorem – Henry May 6 '12 at 16:05

Here is one approach.

Suppose the vertical height of the pyramid is $h$ and the unknown length is $l$. Then looking at the shadow of the the pyramid on the square base you might get something like this

enter image description here

You should be able to find $h$ and then $l$.

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Thank you. How do these help, though? None of these lines are equal (or we don't know it yet) so there's nothing to compare here. Also, we don't know if any of the angles formed by these lines is a right one so how can I use it? – Straightfw May 6 '12 at 10:32

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