Let OABC O is the origin be a tetrahedron with edges length BC -2- CA -3- AB -4- OA -5- OB -6 and OC -7-Then which of the following is are CORRECT-A- Volume of the tetrahedron OABC is -11-4B- Volume of the tetrahedron OABC is -11-4C- Perpendicular distance between the point O and the plane ABC is 3 -11-23D- Perpendicular distance between the point O and the plane ABC is -11-92

The correct options are A Volume of the tetrahedron OABC is √(11)4 C Perpendicular distance between the point O and the plane ABC is 3√(1123)OA=|a|=√(5), OB=|b ...

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Byju's Answer

Standard XII

Mathematics

Area between x=g(y) and y Axis

Let OABC O is...

Question

Let $O A B C$ $(O$ is the origin $)$ be a tetrahedron with edges length $B C = \sqrt{2}$ , $C A = \sqrt{3}$ , $A B = \sqrt{4}$ , $O A = \sqrt{5}$ , $O B = \sqrt{6}$ and $O C = \sqrt{7}$ . Then which of the following is (are) CORRECT?

Volume of the tetrahedron

O A B C

\frac{\sqrt{11}}{4}

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Volume of the tetrahedron

O A B C

\sqrt{\frac{11}{4}}

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Perpendicular distance between the point

O

and the plane

A B C

3 \sqrt{\frac{11}{23}}

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Perpendicular distance between the point

O

and the plane

A B C

\sqrt{\frac{11}{92}}

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Solution

The correct options are
A Volume of the tetrahedron $O A B C$ is $\frac{\sqrt{11}}{4}$
C Perpendicular distance between the point $O$ and the plane $A B C$ is $3 \sqrt{\frac{11}{23}}$
$O A = | \to a | = \sqrt{5}, O B = | \to b | = \sqrt{6}, O C = | \to c | = \sqrt{7}$
$A B = | \to b - \to a | = \sqrt{4} \Rightarrow \to a \cdot \to b = \frac{7}{2}$
$B C = | \to c - \to b | = \sqrt{2} \Rightarrow \to b \cdot \to c = \frac{11}{2}$
$C A = | \to a - \to c | = \sqrt{3} \Rightarrow \to c \cdot \to a = \frac{9}{2}$

$[\to a \to b \to c]^{2} = ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ \begin{matrix} 5 & \frac{7}{2} & \frac{9}{2} \frac{7}{2} & 6 & \frac{11}{2} \frac{9}{2} & \frac{11}{2} & 7 \end{matrix} ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ = \frac{99}{4}$

$∴$ Volume $= \frac{1}{6} [\to a \to b \to c] = \frac{\sqrt{11}}{4}$

Also, Volume $= \frac{1}{3} \times base area \times height$
$= \frac{1}{3} \times ar (△ A B C) \times height \dots (1)$

Now, $ar (△ A B C) = \frac{1}{2} (- - \to A B \times - - \to B C)$
$= \frac{1}{2} A B . B C sin θ$
$= \sqrt{2} sin θ \dots (2)$

$- - \to A B \cdot - - \to B C = | \to b - \to a | | \to c - \to b | cos θ$
$\Rightarrow cos θ = \frac{- - \to A B \cdot - - \to B C}{| \to b - \to a | | \to c - \to b |}$
$\Rightarrow cos θ = \frac{\to b \cdot \to c - \to b \cdot \to b - \to a \cdot \to c + \to a \cdot \to b}{2 \sqrt{2}}$
$\Rightarrow cos θ = \frac{\frac{11}{2} - 6 - \frac{9}{2} + \frac{7}{2}}{2 \sqrt{2}}$
$\Rightarrow cos θ = \frac{- 3}{4 \sqrt{2}}$
$∴ sin θ = \sqrt{1 - {cos}^{2} θ} = \frac{\sqrt{23}}{4 \sqrt{2}} \dots (3)$

From $(1)$ , $(2)$ and $(3)$
Volume $= \frac{1}{3} \times \sqrt{2} \cdot \frac{\sqrt{23}}{4 \sqrt{2}} \times height$
$\Rightarrow \frac{\sqrt{11}}{4} = \frac{\sqrt{23}}{3 \times 4} \times height$
$\Rightarrow height = 3 \sqrt{\frac{11}{23}}$

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Let OABC (O is the origin) be a tetrahedron with edges length BC=√2, CA=√3, AB=√4, OA=√5, OB=√6 and OC=√7. Then which of the following is (are) CORRECT?

Let $O A B C$ $(O$ is the origin $)$ be a tetrahedron with edges length $B C = \sqrt{2}$ , $C A = \sqrt{3}$ , $A B = \sqrt{4}$ , $O A = \sqrt{5}$ , $O B = \sqrt{6}$ and $O C = \sqrt{7}$ . Then which of the following is (are) CORRECT?