# 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 m_{1} : m_{2} i.e.

PR/RQ = m_{1} / m_{2}, where m_{1}
0,
m_{2} 0,
m_{1} + m_{2} 0

Then the coordinates of R are (m_{1} x_{2} +m_{2}
x_{1)/(}m_{1} + m_{2)}, (m_{1}y_{2}
+ m_{2}y_{1)/(}m_{1} + m_{2})

**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 m_{1} : m_{2}
then it divides [QP] in the ratio m_{2} : m_{1}.

### When the Point divides the line segment Externally

Let P (x_{1}, y_{1}) and Q (x_{2}, y_{2})
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 m_{1} : m_{2} i.e.

PR/RQ = m_{1} / m_{2}, where m_{1}
0, m_{2} 0,
m_{1} - m_{2} 0

Then the co-ordinates of R are m_{1} x_{2} -m_{2} x_{1)/(}m_{1}
-m_{2)}, (m_{1}y_{2} -m_{2}y_{1)/(}m_{1} -m_{2})

### Mid-point formula

The co-ordinates of the mid-point of [PQ] are ((x_{1} +x_{2)/2},
(y_{1} +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

- 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) - 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

- 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. - 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. - 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. - Find the co-ordinates of the points of trisection of the line segment joining the points (3, - 1) and (-6, 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.
- Find the point which is one-third of the way from P (3, 1) to Q (-2, 5).
- Find the point which is two third of the way from P(0, 1) to Q(1, 0).
- Find the co-ordinates of the point which is three fifth of the way from (4, 5) to (-1, 0).
- 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.
- In what ratio does the point P (2, -5) divide the line segment joining the points A (- 3, 5) and B (4, -9)?
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- By using section formula, prove that the points (0, 3), (6, 0) and (4, 1) are collinear.
- 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.
- 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.
- 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.
- Show that the line segments joining the points (1, - 2), (1, 2) and (3, 0), (-1, 0) bisect each other.
- Show that the points A(-2, -1), B (1, 0), C (4, 3) and D (1, 2) from a parallelogram. Is it a rectangle?
- 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?
- Three consecutive vertices of a parallelogram are (4, - 11), (5, 3) and (2, 15). Find the fourth vertex.

## Answers

**1.**(i) (3, 1) (ii) (-1, 9)

**2.**(i) (12/5, 13/5) (ii) (0, - 7)

**3.**(i) (4, 8) (ii) (1, - 2)

**4.**(0, 1) and (-3, 3)

**5.**(-1/3, 2/3) and (9, 10)

**6.**(4/3, 7/3)

**7.**(2/3, 1/3)

**8.**(1, 2)

**9.**(4, -11)

**10.**5 : 2 internally

**11.**1 : 2 internally; (3, 0)

**12.**5 : 2 externally; (-1 , 0)

**13.**3 : 2 internally

**14.**(- 14, 6)

**15.**(3, 6)

**16.**3 : 2 internally; (2, 3)

**18.**(11 , 10)

**19.**a = 2, b = 2

**20.**(-7 , 3)

**22.**No

**23.**Parallelogram

**24.**(1, 1)