A ratio compares two quantities, expressed as a:b or a/b. A ratio formula provides the coordinates of various points related to triangles, including the centroid and incentre. The centroid is the intersection of the triangle's medians and divides them in a 2:1 ratio. The incentre is the intersection of the angle bisectors and lies on each angle bisector segment in a ratio determined by the corresponding side lengths.

1. RATIO FORMULA
Mr.Sadiq Hussain.
Mjcc-Ghazi

3. What is a ratio?
• A ratio is a comparison of two quantities
• The ratio of a to b can be expressed as:
a : b
or
a/b
b
aor

4. sssss
Section Formula – Internal Division

6. Midpoint of A(x1, y1) and B(x2,y2)
m:n  1:1
Midpoint

7. 2 1 2 1mx nx my ny
P ,
m n m n
  
     
Section Formula – External Division

8. A(x1, y1)
B(x2, y2) C(x3, y3)D
EF G
2 3 2 3x x y y
D ,
2 2
  
  
 
1 3 1 3x x y y
E ,
2 2
  
  
 
Centroid is always
denoted by G.
1 2 1 2x x y y
F ,
2 2
  
  
 
Intersection of medians of a
triangle is called the centroid
Centroid

9. A(x1, y1)
B(x2, y2) C(x3, y3)D
EF L
1 3 1 3x x y y
E ,
2 2
  
  
 
1 2 1 2x x y y
F ,
2 2
  
  
 
2 3 2 3x x y y
D ,
2 2
  
  
 
Consider points L, M, N dividing AD, BE
and CF respectively in the ratio
2:1
2 3 2 3
1 1
x x y y
x 2 y 2
2 2L ,
1 2 1 2
  
  
  
   
 
Centroid -Explanation

10. A(x1, y1)
B(x2, y2) C(x3, y3)D
EF M
2 3 2 3x x y y
D ,
2 2
  
  
 
1 3 1 3x x y y
E ,
2 2
  
  
 
1 2 1 2x x y y
F ,
2 2
  
  
 
Consider points L, M, N dividing AD, BE
and CF respectively in the ratio 2:1
1 3 1 3
2 2
x x y y
x 2 y 2
2 2M ,
1 2 1 2
  
  
  
   
 
Centroid –contd…….

11. A(x1, y1)
B(x2, y2) C(x3, y3)D
EF N
2 3 2 3x x y y
D ,
2 2
  
  
 
1 3 1 3x x y y
E ,
2 2
  
  
 
1 2 1 2x x y y
F ,
2 2
  
  
 
Consider points L, M, N dividing AD, BE
and CF respectively in the ratio 2:1
1 2 1 2
3 3
x x y y
x 2 y 2
2 2N ,
1 2 1 2
  
  
  
   
 
Centroid –contd…….

12. A(x1, y1)
B(x2, y2) C(x3, y3)D
EF G
1 3 1 3x x y y
E ,
2 2
  
  
 
1 2 1 2x x y y
F ,
2 2
  
  
 
2 3 2 3x x y y
D ,
2 2
  
  
 
1 2 3 1 2 3x x x y y y
L ,
3 3
    
  
 
1 2 3 1 2 3x x x y y y
M ,
3 3
    
  
 
1 2 3 1 2 3x x x y y y
N ,
3 3
    
  
 
We see that L  M  N  G
Medians are
concurrent at the
centroid, centroid
divides medians in
ratio 2:1
Centroid –contd…….

13. A(x1, y1)
B(x2, y2) C(x3, y3)D
EF G
1 3 1 3x x y y
E ,
2 2
  
  
 
1 2 1 2x x y y
F ,
2 2
  
  
 
2 3 2 3x x y y
D ,
2 2
  
  
 
1 2 3 1 2 3x x x y y y
L ,
3 3
    
  
 
1 2 3 1 2 3x x x y y y
M ,
3 3
    
  
 
1 2 3 1 2 3x x x y y y
N ,
3 3
    
  
 
We see that L  M  N  G
Medians are
concurrent at the
centroid, centroid
divides medians in
ratio 2:1
Centroid –contd…….
1 2 3 1 2 3x x x y y y
G ,
3 3
    
  
 

14. Incentre is the
centre of the
incircle
A(x1, y1)
B(x2, y2) C(x3, y3)D
EF I
Let BC = a, AC = b, AB = c
AD, BE and CF are the angle
bisectors of A, B and C respectively.
BD AB b
DC AC c
   2 3 2 3bx cx by cy
D ,
b c b c
  
     
Intersection of angle bisectors of a triangle is
called the incentre
Incentre-point

15. A(x1, y1)
B(x2, y2) C(x3, y3)D
EF I
BD AB b
DC AC c
  
2 3 2 3bx cx by cy
D ,
b c b c
  
     
AI AB AC AB AC c b
Now,
ID BD DC BD DC a
 
   

 
 
 
 
2 3 2 3
1 1
bx cx by cy
ax b c ay b c
b c b cI ,
a b c a b c
  
        
     
 
1 2 3ax bx cx
I
a b c
  
     
Similarly I can be derived using E and
F also
Incentre-contd….

16. A(x1, y1)
B(x2, y2) C(x3, y3)D
EF I
BD AB b
DC AC c
   2 3 2 3bx cx by cy
D ,
b c b c
  
     
AI AB AC AB AC c b
Now,
ID BD DC BD DC a
 
   

 
 
 
 
2 3 2 3
1 1
bx cx by cy
ax b c ay b c
b c b cI ,
a b c a b c
  
        
     
 
1 2 3ax bx cx
I
a b c
  
     
Angle bisectors are concurrent at the
incentre
Incentre-point.