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Question
If two pairs of opposite edges of a tetrahedron are perpendicular, then
If two pairs of opposite edges of a tetrahedron are perpendicular, then
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Solution
The correct option is A The third is also perpendicular
Let OABC be the tetrahedron, where O,A,B and C are (0,0,0),(x1,y1,z1),(x2,y2,z2) and (x3,y3,z3) respectively
The d.r's of OA,OB and OC are x1,y1,z1,x2,y2,z2 and x3,y3,z3 respectively.
Also the d.r's of BC,CA and AB are x3−x2,y3−y2,z3−z2;x1−x3,y1−y3,z1−z3 and x2−x1,y2−y1,z2−z1 respectively.
Let the edge OA be perpendicular to the opposite edge BC
Then x1(x3−x2)+y1(y3−y2)+z1(z3−z2)=0
Also if the edge OB is perpendicular to the opposite edge CA then
x2(x1−x3)+y2(y1−y3)+z2(z1−z3)=0x3(x1−x2)+y3(y1−y2)+z3(z1−z2)=0
which shows that the third pair of opposite edge i.e OC and AB is also perpendicular.
Let OABC be the tetrahedron, where O,A,B and C are (0,0,0),(x1,y1,z1),(x2,y2,z2) and (x3,y3,z3) respectively
The d.r's of OA,OB and OC are x1,y1,z1,x2,y2,z2 and x3,y3,z3 respectively.
Also the d.r's of BC,CA and AB are x3−x2,y3−y2,z3−z2;x1−x3,y1−y3,z1−z3 and x2−x1,y2−y1,z2−z1 respectively.
Let the edge OA be perpendicular to the opposite edge BC
Then x1(x3−x2)+y1(y3−y2)+z1(z3−z2)=0
Also if the edge OB is perpendicular to the opposite edge CA then
x2(x1−x3)+y2(y1−y3)+z2(z1−z3)=0x3(x1−x2)+y3(y1−y2)+z3(z1−z2)=0
which shows that the third pair of opposite edge i.e OC and AB is also perpendicular.
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