NCERT Solutions for Class 11 Maths Exercise 12.3

In 11th Mathematics exercise 12.3 there are five numerical questions and 4 examples preceding that. This exercise is related to the use of section formula which is similar in nature to what we have studied in class 10th while studying two dimensional geometry of coordinate geometry.

In two dimensional geometry we learnt how to find out the coordinates of a point dividing a line segment in a given ratio internally. Here, the formula is extended with respect to three dimension. For example – Let the two given points be P(x₁. y₁, z₁) and Q (x₂, y₂, z₂). Let the point R (x, y, z) dive PQ in the given ratio m : n.

Using the sections formula, we find the coordinates of the points R which divides the line segment joining two points P and Q in ratio m : n internally or externally. Extending this formula we can find out the coordinates of mid-point and we can also find out coordinates of point are which divide PQ in the ratio of 2:1. Collinearity of three given points can also be prove by using the section formula.

The all-important question is example 9 where we are required to find out the coordinate of centroid of the triangle whose vertices are given. The coordinates of centroid G is given by the formula {(x₁+x₂+x₃)/3, (y₁+y₂+y₃)/3, (z₁+z₂+z₃)/3)}.

Recollect that the medians of a triangle are concurrent the centroid divides the medians in the ratio 2:1. Using this concept, we derive the formula for centroid of triangle. This derivation is important for examination point of view. Most of the time it is asked in school exams.

Sometimes question can be asked regarding finding out the coordinates of the point which divides the line segments joining the point P and Q in a particular ratio, externally also, so from examination point of view one must carefully read the question. The last question number 5 is regarding finding the coordinates which trisect the line segment joining the point P and Q.

Again, refresh your memory in you will find that such a question appeared in class 10th NCERT also. Here, the ratio can be 1:2 or 2:1. We must workout both the solutions, unless otherwise said.