(B) The volume $V$ of a tetrahedron with two opposite edges of lengths $a$ and $b$,distance $d$ between them,and angle $\theta$ between them is given

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(B) The volume $V$ of a tetrahedron with two opposite edges of lengths $a$ and $b$,distance $d$ between them,and angle $\theta$ between them is given by $V = \frac{1}{6} abd \sin(\theta)$.Here,$AB = 12$,$CD = 12$,$d = EF = 10$,and since $EF \perp AB$ and $EF \perp CD$,the edges $AB$ and $CD$ are perpendicular to each other,so $\theta = 90^\circ$.Thus,$V = \frac{1}{6} \times 12 \times 12 \times 10 \times \sin(90^\circ)$.$V = \frac{1}{6} \times 144 \times 10 \times 1$.$V = 24 \times 10 = 240$.

Class 12 Mathematics Three Dimensional Geometry Mix Examples-Co-ordinate Geometry of Three Dimensions

Tetrahedron $ABCD$ has side lengths $AB = CD = 12$. These edges are perpendicular to each other. Let $E$ and $F$ be the midpoints of $AB$ and $CD$ respectively. Given that $EF = 10$ and $EF$ is perpendicular to both $AB$ and $CD$,find the volume of the tetrahedron $ABCD$.

A
$200$
B
$240$
C
$280$
D
$320$

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Mix Examples-Co-ordinate Geometry of Three Dimensions

Consider the tetrahedron with the vertices $A(3,2,4)$,$B(x_1, y_1, 0)$,$C(x_2, y_2, 0)$,and $D(x_3, y_3, 0)$. If the triangle $BCD$ is formed by the lines $y=x$,$x+y=6$,and $y=1$,then the centroid of the tetrahedron is

MediumAP EAMCET 2024

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If the orthocentre and the centroid of a triangle are at $(5,2,-6)$ and $(9,6,-4)$ respectively,then its circumcentre is

EasyTS EAMCET 2019

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Find the reflection of the point $(1, 6, 3)$ in the line $\frac{x}{1} = \frac{y - 1}{2} = \frac{z - 2}{3}$.

Difficult

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Find the reflection of the point $(-1, 3, 4)$ in the plane $x - 2y = 0$.

Difficult

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Let $A(1, 8, 4)$, $B(0, -11, 3)$, and $C(2, -3, -1)$ be three points. If $D$ is the foot of the perpendicular from $A$ to the line $BC$, find the coordinates of $D$.

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Tetrahedron $ABCD$ has side lengths $AB = CD = 12$. These edges are perpendicular to each other. Let $E$ and $F$ be the midpoints of $AB$ and $CD$ respectively. Given that $EF = 10$ and $EF$ is perpendicular to both $AB$ and $CD$,find the volume of the tetrahedron $ABCD$.

Similar Questions

Consider the tetrahedron with the vertices $A(3,2,4)$,$B(x_1, y_1, 0)$,$C(x_2, y_2, 0)$,and $D(x_3, y_3, 0)$. If the triangle $BCD$ is formed by the lines $y=x$,$x+y=6$,and $y=1$,then the centroid of the tetrahedron is

If the orthocentre and the centroid of a triangle are at $(5,2,-6)$ and $(9,6,-4)$ respectively,then its circumcentre is

Find the reflection of the point $(1, 6, 3)$ in the line $\frac{x}{1} = \frac{y - 1}{2} = \frac{z - 2}{3}$.

Find the reflection of the point $(-1, 3, 4)$ in the plane $x - 2y = 0$.

Let $A(1, 8, 4)$, $B(0, -11, 3)$, and $C(2, -3, -1)$ be three points. If $D$ is the foot of the perpendicular from $A$ to the line $BC$, find the coordinates of $D$.

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