# Section Formula

Let the two given points be . Let the point divide AC in the given ratio

*internally. Draw AF, CH and BG perpendicular to the XY-plane. Obviously AF || BG || CH and feet of these perpendiculars lie in a XY-plane. The points F, G and H will lie on a line which is the intersection of the plane containing AF, BG and CH with the XY-plane.*

Through the point B draw a line DE parallel to the line FH. Line DE will intersect the line FA externally at the point D and the line HC at E, as shown. Also note that quadrilaterals FGBD and GHEB are parallelograms. The triangles ABD and BCE are similar. Therefore,

Similarly, by drawing perpendiculars to the XZ and YZ-planes, we get

Hence, the coordinates of the point B which divides the line segment joining two points internally in the ratio

# Corollary

- If the point B divides the line AC externally in the ratio
- If the point B is the midpoint of the line AC, then . Thus the coordinates of the point B in this case are .These are the coordinates of the mid point of the line segment joining .
- Suppose B divides AC in the ratio Its coordinates can be got by taking . Thus the coordinates of the point B in this case are .

**Example 1**

Find the coordinates of the point which divides the line segment joining the points in the ratio 2:3 (i) internally, (ii) externally

**Solution:**

Let the required point be

- the coordinates of the point which divides the line segment internally is given by

âˆ´ the required point is

- the coordinates of the point which divides the line segment externally is given by

âˆ´ the required points

**Example 2**

Find the ratio in which the line segment (2, 4, 5) and (3, 5, 4) is divided by YZ - plane

**Solution:**

let the YZ - plane divide the line segment joining in the ration then the coordinates of P are given by

Since P lies on the YZ - plane; its - coordinates is zero

âˆ´ YZ - plane divides

*AB*externally in the ratio 2 : 3

**Example 3**

A point R with -coordinate 4 lies on the line segment joining the points . Find the coordinates of the point R

**Solution:**

Suppose R divides PQ in the ratio The coordinates of the point R are given by

Given the -coordinate of the point R is 4

âˆ´ coordinate

coordinates

coordinate

âˆ´ the coordinates of the point

*R*are