of the triangle. Therefore, it divides the median AD in the ratio : . Hence, the coordinates of G are z z z , , or z z z , , Example Find the ratio in which the line segment joining the points ( , , ) and ( , , – ) is divided by the YZ-plane. Solution Let YZ-plane divides the line segment joining A ( , , ) and B ( , , – ) at P ( x , y , z ) in the ratio k : .
Then the coordinates of P are , , k k k k k k INTRODUCTION TO THREE DIMENSIONAL GEOMETRY Since P lies on the YZ-plane, its x- coordinate is zero, i.e., + = + k k or k = − Therefore, YZ-plane divides AB externally in the ratio : . EXERCISE . . Find the coordinates of the point which divides the line segment joining the points (– , , ) and ( , – , ) in the ratio (i) : internally, (ii) : externally.
. Given that P ( , , – ), Q ( , , – ) and R ( , , – ) are collinear. Find the ratio in which Q divides PR. .
Find the ratio in which the YZ-plane divides the line segment formed by joining the points (– , , ) and ( , – , ). . Using section formula, show that the points A ( , – , ), B (– , , ) and C , , are collinear. .
Find the coordinates of the points which trisect the line segment joining the points P ( , , – ) and Q ( , – , ). Miscellaneous Examples Example Show that the points A ( , , ), B (– , – , – ), C ( , , ) and D ( , , ) are the vertices of a parallelogram ABCD, but it is not a rectangle. Solution To show ABCD is a parallelogram we need to show opposite side are equal Note that. AB = )