(A) Let the length,breadth,and height of the cuboid be $l, b,$ and $h$ respectively.Given the lengths of the face diagonals are $39, 40,$ and $41$:$l^

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(A) Let the length,breadth,and height of the cuboid be $l, b,$ and $h$ respectively.Given the lengths of the face diagonals are $39, 40,$ and $41$:$l^2 + b^2 = 39^2$$b^2 + h^2 = 40^2$$h^2 + l^2 = 41^2$Adding these three equations:$2(l^2 + b^2 + h^2) = 39^2 + 40^2 + 41^2$$2(l^2 + b^2 + h^2) = 1521 + 1600 + 1681$$2(l^2 + b^2 + h^2) = 4802$$l^2 + b^2 + h^2 = 2401$The length of the main diagonal of the cuboid is given by $\sqrt{l^2 + b^2 + h^2}$.Length $= \sqrt{2401} = 49$.

Class 12 Mathematics THREE DIMENSIONAL GEOMETRY Mix Examples-Co-ordinate Geometry of Three Dimensions

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The three different face diagonals of a cuboid (rectangular parallelepiped) have lengths $39, 40, 41$. The length of the main diagonal of the cuboid which joins a pair of opposite corners is

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A
$49$
B
$49 \sqrt{2}$
C
$60$
D
$60 \sqrt{2}$

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The three different face diagonals of a cuboid (rectangular parallelepiped) have lengths $39, 40, 41$. The length of the main diagonal of the cuboid which joins a pair of opposite corners is

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