Rectangular Pyramids
Is it true that $p^2+q^2=s^2+r^2$ when:

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Think about rightangled triangles.
A proof of this result came from students from the Key Stage 3 Maths Club at Strabane Grammar School, Northern Ireland and the following one from Joel, ACS (Barker) Singapore:
(a) In the diagram given (in 2 dimensions) p ^{2} + q ^{2} = r ^{2} + s ^{2}
Proof
Draw 2 lines through V parallel to the edges of the rectangle, dividing the figure into four pairs of rightangled triangles. Label the point on AB as E, on BC as F, on CD as G and on DA as H. By Pythagoras theorem:
p ^{2} = DG ^{2} +DH ^{2} 
q ^{2} = BE ^{2} + BF ^{2} 
r ^{2} = CF ^{2} + CG ^{2} 
s ^{2} = AE ^{2} + AH ^{2} 
Since DG=AE, CG=BE, AH=BF and CF=DH,
p ^{2} + q ^{2} = AE ^{2} + CF ^{2} + CG ^{2} + AH ^{2} = r ^{2} + s ^{2}
(b) If the diagram represents a pyramid on a rectangular base where p, q, r and s are the lengths of the sloping edges then the result p ^{2} + q ^{2} = r ^{2} + s ^{2} still holds true.
Proof
Let V _{1} be the foot of the perpendicular from V to the base ABCD of the pyramid and let h be the height of the pyramid so that VV _{1} = h and let V _{1} A = s _{1} , V _{1} B = q _{1} , V _{1} C = r _{1} , and V _{1} D= p _{1} .
By Pythagoras theorem we have: p _{1} ^{2} + h ^{2} = p ^{2} , q _{1} ^{2} + h ^{2} = q ^{2} , r _{1} ^{2} + h ^{2} = r ^{2} and s _{1} ^{2} + h ^{2} = s ^{2} .
Using the result already proved in 2dimensions, that is
p _{1} ^{2} + q _{1} ^{2} = r _{1} ^{2} + s _{1} ^{2} ,
we get p _{1} ^{2} + q _{1} ^{2} + 2h ^{2} = r _{1} ^{2} + s _{1} ^{2} + 2h ^{2}
so p ^{2} + q ^{2} = r ^{2} + s ^{2} .
Why do this problem?
It provides experience of generalising a result from 2 dimensions to an equivalent result in 3 dimensions. This problem asks the question for them but learners should be encouraged to ask themselves "What if..." and always to think about possible generalisations.Key questions
What comes to mind when a problem involves squares of distances?If we are looking for Pythagoras theorem where are the right angles triangles?