MT_F4_7.1 Divisor of Line Segment_1

Chapter 7 Coordinate Geometry

7.1 Divisor of a Line Segment

A line segment is part of a straight line with two end points with

specific length or distance.
Any point dividing the segment in a particular ratio is known as the

internal point.

  x1  x2 , y1  y2 
2 2
(i) Midpoint

Example: (x1 , y1) (x2 , y2)

   1. Find the coordinates of midpoint of A  2, 2 and B 4, 2 .

   2 4 , 2  (2) 
2 2
Midpoint of CD

  2 , 0 
2 2

 1 , 0

   2. If K  3, 2 is the midpoint of the points M  4, 7 and N , find the

coordinates of the point N .

  x1  x2 , y1  y2 
2 2
Midpoint

Draft the straight line N x, y  3, 2   4  x 7 y 
 2 2 
最好先草图 ˙

 ,

˙K  3, 2  4  x  3 7 y  2
˙M 4, 7 2 2

 4  x  3(2) 7  y  2(2)
 4  x  6 7 y  4

x  2 y  3

 N 2, 3

3. Given that the points P(7, 6), Q(8, 10), R(3, 7)and S are four vertices of a
parallelogram, find the coordinates of the point S.

y ** Parallelogram
- Two pairs of parallel lines
P(7, 6) - Intersection point = midpoint

Q(8, 10)

˙ Midpoint / intersection point
R(3, 7)
x
S(x, y)

0

Midpoint QS = Midpoint PR ** Remember this
important concept

 8  x , 10  y    7  3 , 6  7 
2 2 2 2

 8  x , 10  y   5 , 13  By comparison: 作比较
2 2  2
coordinate-x and coordinate-y

8x  5 10  y  13
2 22

8  x  10 10  y  13
x2
y3
 S2, 3

比例

(ii) Point that internally divides a Line Segment in the ratio m : n

1n ˙  F x2, y2

Gx, y x, y   n x1  m x2 , n y1  m y2 
 m  n m  n 
˙

1m

˙  E x1, y1

Example:

1. The point P internally divides the line segment joining the points

   K 3, 3 and L  2, 4 in the ratio 2 : 5 . Find the coordinates of point P.

L 2, 4 1. Draft the straight line (最好先草图,避免弄错

˙ value of coordinate-x and coordinate-y)
5
˙P 2. Cross multiplication (画两个交叉箭头)
2
3. Formula: Px, y   n x1  m x2 , n y1  m y2 
 m  n m  n 

˙K 3, 3  5(3)  2(2) 5(3)  2(4) 
25 
coordinate P   25 , 
s 

  15  4 , 15  8 
7 7

  11 , 7 
7 7

 1 4 , 1
7

 2. The point Q 1, 9 internally divides the line segment joining the points
 M 7, 6 and N in the ratio1: 3 . Find the coordinates of point N .

˙Nx, y (1, 9)   3(7)  1(x) , 3(6)  1( y) 
 1 3 
3  1 3 

Q(1, 9) (1, 9)   21 x , 18  y 
4 4
1˙
˙M 7, 6 21  x  1 18  y  9
4 4

21  x  4 18  y  36

x  17 y  18

 N17, 18

   3. Point P  1, k divides the line segment joining the points A  3, 2
 and B 2, 8 in the ratio AP : PB . Find the ratio AP : PB and the value of k.

B2, 8 P1, k    n(3)  m(2) , n(2)  m(8) 
 m  n m  n 
˙

n

˙ P(1,k)  3n  2m  1  2n  8m  k
mn mn
m
 3n  2m  (m  n)
˙A 3,2
2m  m  n  3n  2(3)  8(2)  k
23

1. x, y   n x1  m x2 , n y1  m y2  3m  2n 5k  10
 m  n m  n  k 2
m2 m = 2, n = 3
n3
2. AP : PB = m : n m:n  2:3

3. Compare the coordinate-x and

coordinate-y  AP : PB  2 : 3

Text Book page 180

Page 180

**一定要写成分数格式

AP  1
PB 2
AP : PB = 1 : 2

1. (a) P(–3, 4)
(b) P(–2, 1)
(c) P(3, –1)

2. p = –2t
4. (a) AP: PB = 1 : 2, k = –2

(b) AP: PB = 1 : 1, k = 5

3.