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Prove that the lines joining the vertice...

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

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i. Show that the lines joining the vertices of a tetrahedron to the centroids of opposite faces are concurrent. ii. Show that the joins of the midpoints of the opposite edges of a tetrahedron intersect and bisect each other.

The four lines drawing from the vertices of any tetrahedron to the centroid to the centroid of the opposite faces meet in a point whose distance from each vertex is 'k' times the distance from each vertex to the opposite face, where k is

Prove that the perpendicular let fall from the vertices of a triangle to the opposite sides are concurrent.

Prove that the volume of the tetrahedron and that formed by the centroids of the faces are in the ratio of 27:1.

Two coplanar triangles ABC and PQR re such that the lines drawn through the vertices of /_\ABC parallel to the sides of /_\PQR are concurrent. Prove that the lines drawn through the vertices of /_\PQR parallel to sides of /_\ABC are concurrent.

Prove that the line joining the orthocentre to the centroid of a triangle formed by the focal chord of a parabola and tangents drawn at its extremities is parallel to the axis of the parabola.

Prove that the medians of a triangle are concurrent.

Prove that the altitudes of a triangle are concurrent.

Show that the line joining one vertex of a parallelogram to the mid-point of an opposite side trisects the diagonal and is trisected thereat.

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RD SHARMA ENGLISH-ALGEBRA OF VECTORS-All Questions
  1. Find the position vector of a point A in space such that vec O A is i...

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  2. Show that the points A(6,-7,0),B(16 ,-19 ,-4),C(0,3,-6)a n d D(2,-5,1...

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  3. Prove that the lines joining the vertices of a tetrahedron to the ce...

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  4. Find a vector vec r of magnitude 3sqrt(2)u n i t s which makes an an...

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  5. Let vec a= hat i+2 hat ja n d vec b=2 hat i+ hat jdoti s |vec a|=| ve...

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  6. Three vectors of magnitude a ,2a ,3a meet in a point and their directi...

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  7. Show that the vectors vec a-2 vec b+3 vec c , vec a-3 vec b+5 vec ca ...

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  8. Find the angles at which the vector 2 hat i- hat j+2 hat k is inclined...

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  9. Prove that four points 2 vec a+3 vec b- vec c , vec a-2 vec b+3 vec ...

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  10. Find the direction cosines of the vector joining the points A(1,2,-...

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  11. If vec aa n d vec b are two non-collinear vectors, show that points...

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  12. If the position vector vec a of a point (12 ,n) is such that | vec a|...

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  13. If A=(0,1)B=(1,0),C=(1,2),D=(2,1) , prove that vec A B= vec C Ddot

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  14. Show that the points with position vectors vec a-2 vec b+3 vec c ,-2 ...

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  15. Find the position vector of a point R which divides the line joining ...

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  16. If D is the mid-point of the side B C of a triangle A B C , prove that...

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  17. Show that the found points A ,B ,C ,D with position vectors vec a , v...

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  18. Let vec a , vec b , vec c be the position vectors of three distinct p...

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  19. If vec a and vec b are position vectors of points Aa n dB respectivel...

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  20. If vec aa n d vec b are position vectors of Aa n dB respectively, fin...

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