A Note We can also show that ABCD is a parallelogram, using the property that diagonals AC and BD bisect each other. Example Find the equation of the set of the points P such that its distances from the points A ( , , – ) and B (– , , ) are equal. Solution If P ( x , y , z ) be any point such that PA = PB. Now ) ) z z or ) ) z z or x + y – z – = .
Example The centroid of a triangle ABC is at the point ( , , ). If the coordinates of A and B are ( , – , ) and (– , , – ), respectively, find the coordinates of the point C. Solution Let the coordinates of C be ( x , y , z ) and the coordinates of the centroid G be ( , , ). Then x + − , i.e., x = ; y −+ , i.e., y = ; z + , i.e., z = .
Hence, coordinates of C are ( , , ). Miscellaneous Exercise on Chapter . Three vertices of a parallelogram ABCD are A( , – , ), B ( , , – ) and C (– , , ). Find the coordinates of the fourth vertex.
. Find the lengths of the medians of the triangle with vertices A ( , , ), B ( , , ) and ( , , ). . If the origin is the centroid of the triangle PQR with vertices P ( a , , ), Q (– , b , – ) and R( , , c ), then find the values of a , b and c .
. Find the coordinates of a point on y -axis which are at a distance of from the point P ( , – , ). INTRODUCTION TO THREE DIMENSIONAL GEOMETRY . A point R with x -coordinate lies on the line segment joining the points P( , – , ) and Q ( , , ).
Find the coordinates of the point R. [ Hint Suppose R divides PQ in the ratio k : . The coordinates of the point R are given by , , k k k k k ]. .
If A and B be the points ( , , ) and (– , , – ), respectively, find the equation of the set of points P such that PA + PB = k , where k is a constant. Summary