2.Section Formula

Section Formula

Bageecha Singh’s garden is rectangular in shape and its length and breadth are 10 m and 20 m respectively. Two lamp posts in the garden are placed at the ends of a diagonal of the garden. Bageecha Singh wants to place one more lamp post between the two lamp posts that will divide the line segment joining the two lamp posts in the ratio 3:5.

Now, can you help him to find the position of the new lamp post? External division of a line segment:

Observe the figure given below.

Here, AB is a line segment and P is a point outside the line segment AB such that A – B – P (or P – A – B). So, it can be said that the point P divides the line segment AB externally in the ratio AB : BP. Point P is known as the point of external division.

Let the coordinates of points A and B be (x₁, y₁) and (x₂, y₂) respectively. Also, let the point P divide the line segment AB externally in the ratio m : n then the coordinates of point P are given by the following formula.

Coordinates of P =

This formula is known as the section formula for external division.

are the vertices of a ΔABC, then the centroid of ΔABC is

Let us solve some examples based on the section formula.

Example 1: Find the coordinates of the point which divides the line segment joining the points (2, 3) and (–1, 7) internally in the ratio 1:2.

Solution:

Let (2, 3) and (–1, 7) be denoted by A and B respectively.

Let C be the point that divides the line segment AB internally in the ratio 1:2.

Using section formula, we obtain

Coordinates of C

Thus, are the required coordinates of the point.

Example 2: Find the coordinates of the point which divides the line segment joining the points (4, –5) and (6, 2) externally in the ratio 3:2.

Solution:

Let (4, –5) and (6, 2) be denoted by A and B respectively.

Let P be the point that divides the line segment AB externally in the ratio 3:2. Using section formula, we obtain

Coordinates of P

Thus, (10, 16) are the required coordinates of the point.

Example 3: Find the coordinates of the points of trisection of a line segment joining the points (−2, 1) and (4, –3).

Solution:

Let (–2, 1) and (4, –3) be denoted by A and B respectively.

Let C and D be the points of trisection. This means that C divides the line segment AB in the ratio 1 : 2 and D divides the line segment AB in the ratio 2 : 1.

Using section formula, we have

Coordinates of C

and Coordinates of D

Thus, and are the points of trisection of a line segment joining the points (– 2, 1) and (4, –3).

Example 4: The mid-point of a portion of a line that lies in the first quadrant is (3, 2). Find the points at which the line intersects the axes.

Solution:

The line has been shown in the following graph:

Let A and B be the points of intersection with y and x-axes respectively.

Let the coordinates of A and B be (0, b) and (a, 0).

Here, C is the mid-point of A and B.

∴ = (3, 2)

= (3, 2)

On equating the x and y-coordinates on both sides, we obtain

and

∴ a = 6 and b = 4

Thus, the coordinates of A and B are (0, 4) and (6, 0) respectively.