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Standard X

Mathematics

Volume of Regular Pyramids

OA, OB, OC ar...

Question

$O A, O B, O C$ are there mutually perpendicular straight lines in space. If $¯ ¯¯¯¯¯¯ ¯ O A = a, ¯ ¯¯¯¯¯¯ ¯ O B = b$ , and $¯ ¯¯¯¯¯¯ ¯ O C = c$ , then the volume of the pyramid $O A B C$ is $\frac{a b c}{k}$ . Find the value of $k$ .

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Solution

The volume of right triangular prism $=$ area of base $\times$ height
$\frac{a b c}{k} = \frac{1}{2} \times c \times b \times a$

$\frac{a b c}{k} = \frac{a b c}{2}$

$∴$ $k = 2$

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Similar questions

Q. OA perpendicular to OD, OC perpendicular to OB, OD = OA, OC = OB. Then prove that AB = CD

Q. Assertion :In each of the three planes determined by two of the lines

O A, O B, O C

(

O

being the origin), a straight line is drawn through

O

perpendicular to the third line.
The three lines so determined are coplanar. Reason:

(a \times b) \times c + (b \times c) \times a + (c \times a) \times b = 0

, where

O A = a, O B = b

and

O C = c .

Q. Let

- - \to O A = \to a, - - \to O B = 10 \to a + 2 \to b

and

- - \to O C = \to b,

where

O, A

and

C

are non-collinear points. Let

p

denote the area of the quadrilateral

O A B C

, and

q

denote the area of the parallelogram with

O A

and

O C

as adjacent sides. If

p = K q

, then find the value of

K

Q. O is any point in the interior of ∆ABC. Then, which of the following is true?
(a)

(O A + O B + O C) > (A B + B C + C A)

(b)

(O A + O B + O C) > \frac{1}{2} (A B + B C + C A)

(c)

(O A + O B + O C) < \frac{1}{2} (A B + B C + C A)

(d) None of these

Let be three rays such that lies between. Suppose the bisectors of ∠AOC and ∠COB are perpendicular to each other. Prove that B, O, A are collinear.

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OA,OB,OC are there mutually perpendicular straight lines in space. If ¯¯¯¯¯¯¯¯OA=a,¯¯¯¯¯¯¯¯OB=b, and ¯¯¯¯¯¯¯¯OC=c, then the volume of the pyramid OABC is abck. Find the value of k.

The volume of right triangular prism = area of base × height abck=12×c×b×a abck=abc2 ∴ k=2

$O A, O B, O C$ are there mutually perpendicular straight lines in space. If $¯ ¯¯¯¯¯¯ ¯ O A = a, ¯ ¯¯¯¯¯¯ ¯ O B = b$ , and $¯ ¯¯¯¯¯¯ ¯ O C = c$ , then the volume of the pyramid $O A B C$ is $\frac{a b c}{k}$ . Find the value of $k$ .

The volume of right triangular prism $=$ area of base $\times$ height
$\frac{a b c}{k} = \frac{1}{2} \times c \times b \times a$

$\frac{a b c}{k} = \frac{a b c}{2}$

$∴$ $k = 2$