Section Formula

When the Point divides the line segment Internally

Let P (x1, y1) and Q (x2, y2) be two given points in the co-ordinate plane, and R (x, y) be the point which divides the segment [PQ] internally in the ratio m₁ : m₂ i.e.
    PR/RQ = m₁ / m₂, where m₁ 0, m₂ 0, m₁ + m₂ 0

Then the coordinates of R are (m₁ x₂ +m₂ x_1)/(m₁ + m₂₎, (m₁y₂ + m₂y_1)/(m₁ + m₂)

Note. [PQ] stands for the portion of the line PQ which is included between the points P and Q including the points P and Q. [PQ] is called segment directed from P to Q. It may be observed that [QP] is the segment directed from Q to P. If a point R divides [PQ] in the ratio m₁ : m₂ then it divides [QP] in the ratio m₂ : m₁.

When the Point divides the line segment Externally

Let P (x₁, y₁) and Q (x₂, y₂) be two given points in the co-ordinate plane, and R (x, y) be the point which divides the segment [PQ] externally in the ratio m₁ : m₂ i.e.
     PR/RQ = m₁ / m₂, where m₁ 0, m₂ 0, m₁ - m₂ 0
Then the co-ordinates of R are m₁ x₂ -m₂ x_1)/(m₁ -m₂₎, (m₁y₂ -m₂y_1)/(m₁ -m₂)

Mid-point formula

The co-ordinates of the mid-point of [PQ] are ((x₁ +x_2)/2, (y₁ +y_2)/2)

Illustrative Examples

Example

Find the co-ordinates of the point which divides the line segment joining the points P (2, -3) and Q (-4, 5) in the ratio 2 : 3
(i) internally
(ii) externally.
                                            

Solution

1. Let (x, y) be the co-ordinates of the point R which divides the line segment joining the points P (2, -3) and Q (-4, 5) in the ratio 2 : 3 internally, then
   x = [2.(-4) +3.2]/(2+3) = - 2/5 and
   y = [2.5 +3.(-3)]/(2+3) = 1/5
   Hence the co-ordinates of R are (-2/5, 1/5)
2. Let (x, y) be the co-ordinates of the point R which divides the line segment joining the points P (2, - 3) and Q (-4, 5) in the ratio 2 : 3 externally i.e.internally in the ratio 2 : -3.
                               
   x = [2.(-4) + (-3).2]/[2 +(-3)] = -14/1 = 14
   and y = [2.5 + (-3)(-3)]/[2 +(-3)] = 19/(-1) -19
   Hence the co-ordinates of R are (14, -19).

Example

In what ratio is the line segment joining the points (4, 5) and (1, 2) divided by the y-axis? Also find the co-ordinates of the point of division.

Solution

Let the line segment joining the points A (4, 5) and B (1, 2) be divided by the y-axis in the ratio k : 1 at P.
By section formula, co-ordinates of P are ((k +4)/(k+1), (2k +5)/(k+1)).
But P lies on y-axis, therefore, x-coordinate of P = 0
=>    (k +4)/(k+1) = 0   =>     k +4 = 0      =>    k = -4
The required ratio is -4 : 1 or 4 : 1 externally.
Also the co-ordinates of the point of division are
    (0, (2.(-4) +5)/(-4+1)) i.e (0, 1)

Exercise

1. Find the co-ordinates of the point which divides the join of the points (2, 3) and (5, -3) in the ratio 1 : 2
   (i) internally
   (ii) externally.
2. Find the co-ordinates of the point which divides the join of the points (2, 1) and (3, 5) in the ratio 2 : 3
   (i) internally
   (ii) externally.
3. Find the co-ordinates of the point that divides the segment [PQ] in the given ratio:
   (i) P (5, -2), Q (9, 6) and ratio 3 : 1 internally.
   (ii) P (-7, 2), Q (-1, -1) and ratio 4 : 1 externally.
4. Find the co-ordinates of the points of trisection of the line segment joining the points (3, - 1) and (-6, 5).
5. Find point (or points) on the line through A (- 5, -4) and B (2, 3) that is twice as far from A as from B.
6. Find the point which is one-third of the way from P (3, 1) to Q (-2, 5).
7. Find the point which is two third of the way from P(0, 1) to Q(1, 0).
8. Find the co-ordinates of the point which is three fifth of the way from (4, 5) to (-1, 0).
9. If P (1, 1) and Q (2, -3) are two points and R is a point on PQ produced such that PR = 3 PQ, find the co-ordinates of R.
10. In what ratio does the point P (2, -5) divide the line segment joining the points A (- 3, 5) and B (4, -9)?
11. In what ratio is the line joining the points (2, - 3) and (5, 6) divided by the x-axis? Also find the co-ordinates of the point of division.
12. In what ratio is the line joining the points (4, 5) and (1, 2) divided by the x-axis? Also find the co-ordinates of the point of division.
13. In what ratio is the line joining the points (3, 4) and (- 2, 1) divided by the y-axis? Also find the co-ordinates of the point of division.
14. Point C (-4, 1) divides the line segment joining the points A (2, - 2) and B in the ratio 3 : 5. Find the point B.
15. The point R (-1, 2) divides the line segment joining P (2, 5) and Q in the ratio 3 : 4 externally, find the point Q.
16. Find the ratio in which the point P whose ordinate is 3 divides the join of (-4, 3) and (6, 3), and hence find the co-ordinates of P.
17. By using section formula, prove that the points (0, 3), (6, 0) and (4, 1) are collinear.
18. Points P, Q, R are collinear. The co-ordinates of P, Q are (3, 4), (7, 7) respectively and length PR = 10 unit, find the co-ordinates of R.
19. The mid-point of the line segment joining (2 a, 4) and (-2, 3 b) is (1, 2 a +1). Find the values of a and b.
20. The center of a circle is (-1, 6) and one end of a diameter is (5, 9), find the co-ordinates of the other end.
21. Show that the line segments joining the points (1, - 2), (1, 2) and (3, 0), (-1, 0) bisect each other.
22. Show that the points A(-2, -1), B (1, 0), C (4, 3) and D (1, 2) from a parallelogram. Is it a rectangle?
23. The vertices of a quadrilateral are (1, 4), (- 2, 1), (0, -1) and (3, 2). Show that the diagonals bisect each other. What does quadrilateral become?
24. Three consecutive vertices of a parallelogram are (4, - 11), (5, 3) and (2, 15). Find the fourth vertex.

Answers