If K is the length of any edge of a regular tetrahedron, then the distance of any vertex from the opposite face is

The length of the side of a transparent cube is 18 cm . Inside the cube there is an air bubble. If it is seen from one face of the cube, the depth of the bubble is 2 cm and if it is seen from the opposite face the depth is 10 cm. What is the refractive index of the cube and what is the distance of the bubble from the first face?

The ratio of the sides of a triangle is 3:4:5 and its area is 7776 sq-cm. Find the length of the sides of the triangle and also find the perpendicular distance of its greates sides from the oppsite vertex of it.

The lengths of two opposite edges of a tetrahedron are a and b ; the shortest distane between these edges is d , and the angel between them is theta Prove using vectors that the volume of the tetrahedron is (a b dsi ntheta)/6 .

The distance between two parallel lines is unity. A point P lies between the lines at a distance a from one of them. Find the length of a side of an equilateral triangle PQR, vertex Q of which lies on one of the parallel lines and vertex R lies on the other line.

If the length of a focal chord of the parabola y^2=4a x at a distance b from the vertex is c , then prove that b^2c=4a^3 .

Prove that the lines joining the vertices of a tetrahedron to the centroids of opposite faces are concurrent.

A transparent cube of 0.21m edge contains a small air bubble. Its apparent distances when viewed through one face of the cube is 0.1 m and when viewed from the opposite face is 0.04 m. The actual distance of the bubble from the second face of the cube is

The position vectors of the four angular points of a tetrahedron OABC are (0, 0, 0); (0, 0,2) , (0, 4,0) and (6, 0, 0) respectively. A point P inside the tetrahedron is at the same distance r from the four plane faces of the tetrahedron. Find the value of r

If V be the volume of a tetrahedron and V ' be the volume of another tetrahedran formed by the centroids of faces of the previous tetrahedron and V=K V^(prime),t h e nK is equal to a. 9 b. 12 c. 27 d. 81