(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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THREE DIMENSIONAL GEOMETRY

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Mix Examples-Co-ordinate Geometry of Three Dimensions

Find the equation of the set of points $P$,the sum of whose distances from $A(4, 0, 0)$ and $B(-4, 0, 0)$ is equal to $10$.

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If $A=(2,3,4)$ and $B=(-2,3,4)$,then the locus of a point $P(x,y,z)$ such that $PA+PB=4$ is

DifficultAP EAMCET 2023

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Let $A(1, 2, 3)$,$B(-1, 4, 6)$,$C(0, -6, 4)$,and $D(1, 1, 1)$ be the vertices of a tetrahedron. Let $G$ be its centroid and $G_1$ be the centroid of its face $BCD$. Then $\frac{AG_1}{AG} =$

MediumTS EAMCET 2022

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The point in the $xy$-plane which is equidistant from the points $A(2,0,3)$,$B(0,3,2)$,and $C(0,0,1)$ has the coordinates

MediumAP EAMCET 2025

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Find the length of the perpendicular drawn from the point $P(2\hat{i} - \hat{j} + 5\hat{k})$ to the line $\vec{r} = (11\hat{i} - 2\hat{j} - 8\hat{k}) + \lambda(10\hat{i} - 4\hat{j} - 11\hat{k})$.

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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

Find the equation of the set of points $P$,the sum of whose distances from $A(4, 0, 0)$ and $B(-4, 0, 0)$ is equal to $10$.

If $A=(2,3,4)$ and $B=(-2,3,4)$,then the locus of a point $P(x,y,z)$ such that $PA+PB=4$ is

Let $A(1, 2, 3)$,$B(-1, 4, 6)$,$C(0, -6, 4)$,and $D(1, 1, 1)$ be the vertices of a tetrahedron. Let $G$ be its centroid and $G_1$ be the centroid of its face $BCD$. Then $\frac{AG_1}{AG} =$

The point in the $xy$-plane which is equidistant from the points $A(2,0,3)$,$B(0,3,2)$,and $C(0,0,1)$ has the coordinates

Find the length of the perpendicular drawn from the point $P(2\hat{i} - \hat{j} + 5\hat{k})$ to the line $\vec{r} = (11\hat{i} - 2\hat{j} - 8\hat{k}) + \lambda(10\hat{i} - 4\hat{j} - 11\hat{k})$.

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