Section Formulae

Simply, section formulae refer to the external and internal division of a line segment by a given point.
Section formulae have two types. They are,

1. Section formulae for an internal division.
2. Section formulae for an external division.

Section Formulae for Internal Division
Let's take a line with two ends point A(x1, y2) and B(x2, y2) which are joined by the line segment AB. Consider P(x, y) be any point on AB which divides the line internally in the ratio m1:m2
i.e. AP:PB = m1:m2
The formula for the section formulae in internal division is (x, y) = ($$\frac{m_1x_2+m_2x_1}{m_1+m_2}$$),($$\frac{m_1 y_2 + m_2 y_1}{m_1 + m_2}$$)

Section Formulae for External Division
If the point P(x, y) divides AB externally in the ratio of m1:m2 then the divided segment BP is measured in opposite direction and hence m2 is taken as negative.

$$\therefore$$ The section formulae for external division is,
(x, y) = ($$\frac{m_1x_2 - m_2x_1}{m_1 - m_2}$$),($$\frac{m_1y_2 - m_2y_1}{m_1 - m_2}$$)

In special case, the midpoint formulae is also used'
m1:m2 = 1:1 i.e. m1 = m2
$$\therefore$$ x = $$\frac{x_1 + x_2}{2}$$ and y = $$\frac{y_1 + y_2}{2}$$
Thus, co-ordinates P(x, y) are P($$\frac{x_1 + x_2}{2}$$,$$\frac{y_1 + y_2}{2}$$) which is called mid-point formulae.

• Internal division(x,y) = ($$\frac{m_1x_1+m_2x_1}{m_1+m_2}$$), ($$\frac{m_1y_1+m_2y_1}{m_1+m_2}$$)
• External division(x,y) = ($$\frac{m_1x_1-m_2x_1}{m_1-m_2}$$), ($$\frac{m_1y_1-m_2y_1}{m_1-m_2}$$)
• Mid-point formulae = ($$\frac{x_1+x_2}{2}$$, $$\frac{y_1+y_2}{2}$$)
0%

(5, 5)
(3, 5)
(3, 3)
(3, 2)

4, 1
2, 3
5, 3
3, 5

2 : 3
6 : 4
3 : 5
2 : 5

3 : 9
2 : 3
3 : 5
2 : 9
• Find the co-ordinates of the mid point of the line segment joining the given points (5a, 7b) and (3a, −2b).

(8, (frac{5b}{2}))
(4, (frac{5b}{2}))
(4, (frac{6b}{4}))
(3, (frac{5b}{3}))

0, 0
2, 3
1, 2
2, 1

-6, 0
-6, -0
-1, -6
0, -6

-3, -1
-2, 2,
-2, -1
1, 2

-4, -5
4, 5
4, -5
-5, 4
• Find the co-ordinates of the point which divides the line segment joining (2, 2) and (-2, -2) points in the 2:5 ratios.

((frac{6}{7}), (frac{6}{7}))
((frac{5}{7}), (frac{6}{5}))
2 ,6
5, 6

1, 3
3, 3
1, 2
2, 3

4, 5
-4, 5
-5, 4
-4, -5

3, 4
-3, -4
-4, -3
-3, 4

3, 0
0, 3
-3, 0
1, -2

2, 3
3, 5
5, 3
-3, -2