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Question

If in a tetrahedron, edges in each of the two pairs of opposite edges are perpendicular, then show that the edges in the third pair are also perpendicular.

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Solution

Let ABCD is a tetrahedron and AD,

BC and AC, BD are two pairs of opposite edges which are perpendicular to each other

I.e Ad.Bd = 0 amd AC.Bd and OD = d

Now AD.BC = 0

(da)(cb)=0 ...(1)

and Ac.Bd = 0

(ca)(db)=0 ...(2)

Now equation (1) and (2) we get

(d-a) (c-d) - (c-a) (d-b) = 0

(dcdbac+ab)(cdcb+cd+ab)=0

cddbac+abcd+cbadab=0

cdadabac=0

cbdbadac=0

b(cd)+a(dc)=0

b(cd)a(cd)=0

(cd)(ba)=0

CD.AB=0

Hence, the third pair of opposite edges AB,CD are also perpendicular to each other.





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