Section Formula

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 formula for internal division

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.

.
Section formula for external division

\(\therefore\) 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%
  • Find the co-ordinates of the mid point of the line segement joining the given points, (4, 6) and (2, 4).

    (5, 5)
    (3, 5)
    (3, 3)
    (3, 2)
  • Find the co-ordinates of the point (2, 3) and (7, 3) which divides the line segment joining the following points in the 3:2 ratio.

    4, 1
    2, 3
    5, 3
    3, 5
  • Find the ratio in which the y-axis divides the line segment joining the points (−4, 1) and (10, 1)

    2 : 3
    6 : 4
    3 : 5
    2 : 5
  • Find the ratio in which x-axis divides the line segment joining the points (3, 2) and (3, −9).

    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}))
  • Find the co-ordinates of the point which divides the line segment joining the (2, −4) and (−3, 6) points in the 2:3 ratio.

    0, 0
    2, 3
    1, 2
    2, 1
  • Find the co-ordinates of the point which divides the line segment joining (-10, 12) and (-3, -9) points in the 4:3 ratio.

    -6, 0
    -6, -0
    -1, -6
    0, -6
  • Find the co-ordinates of the mid point of the line segment joining the (8, 5) and (-12, -7) points.

    -3, -1
    -2, 2, 
    -2, -1
    1, 2
  • Find the co-ordinates of the mid point of the line segment joining (-3, -7) and (-5, -3) points.

    -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
  • Find co-ordinates: (1, -3) and (4, 6), ratio 2:1

    1, 3
    3, 3
    1, 2
    2, 3
  • Find co-ordinates: (-3, -7) and (-5, -3)

    4, 5
    -4, 5
    -5, 4
    -4, -5
  • Find co-ordinates: (-6, 0) and (0, -8)

    3, 4
    -3, -4
    -4, -3
    -3, 4
  • Find co-ordinates: (-6, 8) and (6, -2)

    3, 0
    0, 3
    -3, 0
    1, -2
  • Find co-ordinates: (4, 6) and (2, 4)

    2, 3
    3, 5
    5, 3
    -3, -2
  • You scored /15


    Take test again

Any Questions on Section Formula ?

Please Wait...

Discussions about this note

Forum Time Replies Report

pankaj

Show that the points A (3,4) B(7,3) and C (4,8) are the vertices of an isosceles