Let L,M,N be the feet of the perpendiculars drawn from a point P(3,4,5) on the X,Y and Z-axes respectively.Find the coordinates of L, M and N.

Find the centroid of a triangle, the mid-point of whose sides are D(1, 2, -3), E (3, 0, 1) and F(-1, 1, -4).

Find the point on X-axis which is equidistant from the points A(3,2,2) and B(5,5,4)

Find the equation of the set of points which are equidistant from the points (1, 2, 3) and (3, 2, –1).

If origin is the centroid of a \(\Delta ABC\) with vertices \(A(\alpha ,1,3) \ B(-2,\beta ,-5) \ and \ C(4,7,\gamma )\), then find the values of \(\alpha ,\beta \ and \ \gamma \) .

Three students are standing in a park with signboards "SAVE ENVIRONMENT". "DONT'T LITTER", "KEEP YOUR PLACE CLEAN". Their positions are marked by the points A(0, 7, 10), B(-1, 6, 6) and C(-4, 9, 6). The three students are holding GREEN coloured ribbon together. Does the ribbons form sides of a right angled triangle? Do you feel the need to promote? What message is given from this question to the society?

Find the coordinates of a point equidistant from the four points O(0,0,0), A(l,0,0), B(0,m,0), and C(0,0,n)

In the given figure, if the coordinates of point P are (a,b,c), then write the coordinates of A,D,B,C and E.

Show that the points (0,7,10), (-1,6,6) and (-4,9,6) are the vertices of a right angled isosceles triangle.

Prove that the points (0, -1, -7), (2, 1, -9) and (6, 5, -13) are collinear. Find the ratio in which the first point divides the join of the other two.

Show that \(\Delta ABC\) with vertices A(0,4,1), B(2,3,-1), and C(4,5,0) is right angled.

Three points A(1,2,3), B(0,4,1) and C(-1,-1,-3) are the vertices of \(\Delta ABC\). Find the point in which the bisector of \(\angle BAC\) meets BC.

Show that the points (-1,-6,10), (1,-3,4) (-5,-1,1) and (-7,-4,7) are the vertices of a rhombus.

Verify that (-1, 2, 1), (1, -2, 5), (4, -7, 8) and (2, -3, 4) are the vertices of a parallelogram.
Tp prove a quadrilateral is a parallelogram, we have to prove that its diagonal bisect each other

Three points A(3, 2, 0), B(5, 3, 2) and C(-9, 6, -3) are forming a triangle . The bisector Ad of