6. Coordinate Geometry
Distance Formula
( x2 − x1 ) + ( y2 − y1 )
2 2
d=
e.g. Find the distance between
(–1,3) and (3,5)
( 5 − 3) + ( 3 + 1)
2 2
d=
= 22 + 42
= 20
= 2 5 units
7. Coordinate Geometry
Distance Formula
( x2 − x1 ) + ( y2 − y1 )
2 2
d=
e.g. Find the distance between
(–1,3) and (3,5)
( 5 − 3) + ( 3 + 1)
2 2
d=
= 22 + 42
= 20
= 2 5 units
The distance formula is
finding the length of the hypotenuse,
using Pythagoras
8. Coordinate Geometry
Distance Formula Midpoint Formula
( x2 − x1 ) + ( y2 − y1 )
2 2
d=
e.g. Find the distance between
(–1,3) and (3,5)
( 5 − 3) + ( 3 + 1)
2 2
d=
= 22 + 42
= 20
= 2 5 units
The distance formula is
finding the length of the hypotenuse,
using Pythagoras
9. Coordinate Geometry
Distance Formula Midpoint Formula
x1 + x2 y1 + y2
M =
( x2 − x1 ) + ( y2 − y1 )
2 2
,
d=
2
2
e.g. Find the distance between
(–1,3) and (3,5)
( 5 − 3) + ( 3 + 1)
2 2
d=
= 22 + 42
= 20
= 2 5 units
The distance formula is
finding the length of the hypotenuse,
using Pythagoras
10. Coordinate Geometry
Distance Formula Midpoint Formula
x1 + x2 y1 + y2
M =
( x2 − x1 ) + ( y2 − y1 )
2 2
,
d=
2
2
e.g. Find the distance between e.g. Find the midpoint of
(–1,3) and (3,5) (3,4) and (–2 ,1)
( 5 − 3) + ( 3 + 1)
2 2
d=
= 22 + 42
= 20
= 2 5 units
The distance formula is
finding the length of the hypotenuse,
using Pythagoras
11. Coordinate Geometry
Distance Formula Midpoint Formula
x1 + x2 y1 + y2
M =
( x2 − x1 ) + ( y2 − y1 )
2 2
,
d=
2
2
e.g. Find the distance between e.g. Find the midpoint of
(–1,3) and (3,5) (3,4) and (–2 ,1)
3 − 2 4 +1
( 5 − 3) + ( 3 + 1)
2 2
d= M = ,
2 2
= 22 + 42
= 20
= 2 5 units
The distance formula is
finding the length of the hypotenuse,
using Pythagoras
12. Coordinate Geometry
Distance Formula Midpoint Formula
x1 + x2 y1 + y2
M =
( x2 − x1 ) + ( y2 − y1 )
2 2
,
d=
2
2
e.g. Find the distance between e.g. Find the midpoint of
(–1,3) and (3,5) (3,4) and (–2 ,1)
3 − 2 4 +1
( 5 − 3) + ( 3 + 1)
2 2
d= M = ,
2 2
= 22 + 42
= ,
15
= 20 2 2
= 2 5 units
The distance formula is
finding the length of the hypotenuse,
using Pythagoras
13. Coordinate Geometry
Distance Formula Midpoint Formula
x1 + x2 y1 + y2
M =
( x2 − x1 ) + ( y2 − y1 )
2 2
,
d=
2
2
e.g. Find the distance between e.g. Find the midpoint of
(–1,3) and (3,5) (3,4) and (–2 ,1)
3 − 2 4 +1
( 5 − 3) + ( 3 + 1)
2 2
d= M = ,
2 2
= 22 + 42
= ,
15
= 20 2 2
= 2 5 units
The midpoint formula
The distance formula is
averages the x and y values
finding the length of the hypotenuse,
using Pythagoras
15. Division Of An Interval
Mathematics (2 unit) division of an interval questions are restricted
to midpoint questions i.e. dividing in the ratio 1:1
16. Division Of An Interval
Mathematics (2 unit) division of an interval questions are restricted
to midpoint questions i.e. dividing in the ratio 1:1
In Extension 1 you can be asked to divide an interval in a any ratio,
and it could be either an internal or an external division.
17. Division Of An Interval
Mathematics (2 unit) division of an interval questions are restricted
to midpoint questions i.e. dividing in the ratio 1:1
In Extension 1 you can be asked to divide an interval in a any ratio,
and it could be either an internal or an external division.
X Y
18. Division Of An Interval
Mathematics (2 unit) division of an interval questions are restricted
to midpoint questions i.e. dividing in the ratio 1:1
In Extension 1 you can be asked to divide an interval in a any ratio,
and it could be either an internal or an external division.
X A Y
19. Division Of An Interval
Mathematics (2 unit) division of an interval questions are restricted
to midpoint questions i.e. dividing in the ratio 1:1
In Extension 1 you can be asked to divide an interval in a any ratio,
and it could be either an internal or an external division.
A divides XY internally
in the ratio m:n
X A Y
20. Division Of An Interval
Mathematics (2 unit) division of an interval questions are restricted
to midpoint questions i.e. dividing in the ratio 1:1
In Extension 1 you can be asked to divide an interval in a any ratio,
and it could be either an internal or an external division.
A divides XY internally
in the ratio m:n
X A Y
m n
21. Division Of An Interval
Mathematics (2 unit) division of an interval questions are restricted
to midpoint questions i.e. dividing in the ratio 1:1
In Extension 1 you can be asked to divide an interval in a any ratio,
and it could be either an internal or an external division.
A divides XY internally
in the ratio m:n
OR
X A Y
m n
22. Division Of An Interval
Mathematics (2 unit) division of an interval questions are restricted
to midpoint questions i.e. dividing in the ratio 1:1
In Extension 1 you can be asked to divide an interval in a any ratio,
and it could be either an internal or an external division.
A divides XY internally
in the ratio m:n
OR
X A Y
A divides YX internally
m n
in the ratio n:m
23. Division Of An Interval
Mathematics (2 unit) division of an interval questions are restricted
to midpoint questions i.e. dividing in the ratio 1:1
In Extension 1 you can be asked to divide an interval in a any ratio,
and it could be either an internal or an external division.
A divides XY internally
in the ratio m:n
OR
X A Y
A divides YX internally
m n
in the ratio n:m
X Y
24. Division Of An Interval
Mathematics (2 unit) division of an interval questions are restricted
to midpoint questions i.e. dividing in the ratio 1:1
In Extension 1 you can be asked to divide an interval in a any ratio,
and it could be either an internal or an external division.
A divides XY internally
in the ratio m:n
OR
X A Y
A divides YX internally
m n
in the ratio n:m
X Y A
25. Division Of An Interval
Mathematics (2 unit) division of an interval questions are restricted
to midpoint questions i.e. dividing in the ratio 1:1
In Extension 1 you can be asked to divide an interval in a any ratio,
and it could be either an internal or an external division.
A divides XY internally
in the ratio m:n
OR
X A Y
A divides YX internally
m n
in the ratio n:m
A divides XY externally
X Y A in the ratio m:n
26. Division Of An Interval
Mathematics (2 unit) division of an interval questions are restricted
to midpoint questions i.e. dividing in the ratio 1:1
In Extension 1 you can be asked to divide an interval in a any ratio,
and it could be either an internal or an external division.
A divides XY internally
in the ratio m:n
OR
X A Y
A divides YX internally
m n
in the ratio n:m
n
A divides XY externally
X Y A in the ratio m:n
m
27. Type 1: Internal Division 2003 Extension 1 HSC Q1c)
( − 3,4)and
Find the coordinates of P that divides the interval joining
( 5,6) internally in the ratio 1 : 3
28. Type 1: Internal Division 2003 Extension 1 HSC Q1c)
Find the coordinates of P that divides the interval joining ( − 3,4 )and
( 5,6) internally in the ratio 1 : 3
• Write down the endpoints of the interval in the same order as
they are mentioned.
29. Type 1: Internal Division 2003 Extension 1 HSC Q1c)
Find the coordinates of P that divides the interval joining ( − 3,4 )and
( 5,6) internally in the ratio 1 : 3
• Write down the endpoints of the interval in the same order as
they are mentioned.
( − 3,4) ( 5,6)
30. Type 1: Internal Division 2003 Extension 1 HSC Q1c)
Find the coordinates of P that divides the interval joining ( − 3,4 )and
( 5,6) internally in the ratio 1 : 3
• Write down the endpoints of the interval in the same order as
they are mentioned.
• Write down the ratio.
( − 3,4) ( 5,6)
31. Type 1: Internal Division 2003 Extension 1 HSC Q1c)
Find the coordinates of P that divides the interval joining ( − 3,4 )and
( 5,6) internally in the ratio 1 : 3
• Write down the endpoints of the interval in the same order as
they are mentioned.
• Write down the ratio.
( − 3,4) ( 5,6)
1: 3
32. Type 1: Internal Division 2003 Extension 1 HSC Q1c)
Find the coordinates of P that divides the interval joining ( − 3,4 )and
( 5,6) internally in the ratio 1 : 3
• Write down the endpoints of the interval in the same order as
they are mentioned.
• Write down the ratio.
• Draw a cross joining the ratio to the two points
( − 3,4) ( 5,6)
1: 3
33. Type 1: Internal Division 2003 Extension 1 HSC Q1c)
Find the coordinates of P that divides the interval joining ( − 3,4 )and
( 5,6) internally in the ratio 1 : 3
• Write down the endpoints of the interval in the same order as
they are mentioned.
• Write down the ratio.
• Draw a cross joining the ratio to the two points
( − 3,4) ( 5,6)
1: 3
34. Type 1: Internal Division 2003 Extension 1 HSC Q1c)
Find the coordinates of P that divides the interval joining ( − 3,4 )and
( 5,6) internally in the ratio 1 : 3
• Write down the endpoints of the interval in the same order as
they are mentioned.
• Write down the ratio.
• Draw a cross joining the ratio to the two points
• Set up your answer by drawing a set of parentheses with two
vinculums separated by a comma
( − 3,4) ( 5,6)
1: 3
35. Type 1: Internal Division 2003 Extension 1 HSC Q1c)
Find the coordinates of P that divides the interval joining ( − 3,4 )and
( 5,6) internally in the ratio 1 : 3
• Write down the endpoints of the interval in the same order as
they are mentioned.
• Write down the ratio.
• Draw a cross joining the ratio to the two points
• Set up your answer by drawing a set of parentheses with two
vinculums separated by a comma
( − 3,4) ( 5,6) P=
,
1: 3
36. Type 1: Internal Division 2003 Extension 1 HSC Q1c)
Find the coordinates of P that divides the interval joining ( − 3,4 )and
( 5,6) internally in the ratio 1 : 3
• Write down the endpoints of the interval in the same order as
they are mentioned.
• Write down the ratio.
• Draw a cross joining the ratio to the two points
• Set up your answer by drawing a set of parentheses with two
vinculums separated by a comma
• Add the numbers in the ratio together to get the denominator
( − 3,4) ( 5,6) P=
,
1: 3
37. Type 1: Internal Division 2003 Extension 1 HSC Q1c)
Find the coordinates of P that divides the interval joining ( − 3,4 )and
( 5,6) internally in the ratio 1 : 3
• Write down the endpoints of the interval in the same order as
they are mentioned.
• Write down the ratio.
• Draw a cross joining the ratio to the two points
• Set up your answer by drawing a set of parentheses with two
vinculums separated by a comma
• Add the numbers in the ratio together to get the denominator
( − 3,4) ( 5,6) P=
,
4 4
1: 3
38. Type 1: Internal Division 2003 Extension 1 HSC Q1c)
Find the coordinates of P that divides the interval joining ( − 3,4 )and
( 5,6) internally in the ratio 1 : 3
• Write down the endpoints of the interval in the same order as
they are mentioned.
• Write down the ratio.
• Draw a cross joining the ratio to the two points
• Set up your answer by drawing a set of parentheses with two
vinculums separated by a comma
• Add the numbers in the ratio together to get the denominator
• Multiply along the cross and add to get the numerator
( 5,6) P =
( − 3,4)
,
4 4
1: 3
39. Type 1: Internal Division 2003 Extension 1 HSC Q1c)
Find the coordinates of P that divides the interval joining ( − 3,4 )and
( 5,6) internally in the ratio 1 : 3
• Write down the endpoints of the interval in the same order as
they are mentioned.
• Write down the ratio.
• Draw a cross joining the ratio to the two points
• Set up your answer by drawing a set of parentheses with two
vinculums separated by a comma
• Add the numbers in the ratio together to get the denominator
• Multiply along the cross and add to get the numerator
( 5,6) P = 3 × −3 + 1× 5 ,
( − 3,4)
4 4
1: 3
40. Type 1: Internal Division 2003 Extension 1 HSC Q1c)
Find the coordinates of P that divides the interval joining ( − 3,4 )and
( 5,6) internally in the ratio 1 : 3
• Write down the endpoints of the interval in the same order as
they are mentioned.
• Write down the ratio.
• Draw a cross joining the ratio to the two points
• Set up your answer by drawing a set of parentheses with two
vinculums separated by a comma
• Add the numbers in the ratio together to get the denominator
• Multiply along the cross and add to get the numerator
( 5,6) P = 3 × −3 + 1× 5 , 3 × 4 + 1× 6
( − 3,4)
4 4
1: 3
41. Type 1: Internal Division 2003 Extension 1 HSC Q1c)
Find the coordinates of P that divides the interval joining ( − 3,4 )and
( 5,6) internally in the ratio 1 : 3
• Write down the endpoints of the interval in the same order as
they are mentioned.
• Write down the ratio.
• Draw a cross joining the ratio to the two points
• Set up your answer by drawing a set of parentheses with two
vinculums separated by a comma
• Add the numbers in the ratio together to get the denominator
• Multiply along the cross and add to get the numerator
( 5,6) P = 3 × −3 + 1× 5 , 3 × 4 + 1× 6
( − 3,4)
4 4
− 4 18
= ,
1: 3 4 4
42. Type 1: Internal Division 2003 Extension 1 HSC Q1c)
Find the coordinates of P that divides the interval joining ( − 3,4 )and
( 5,6) internally in the ratio 1 : 3
• Write down the endpoints of the interval in the same order as
they are mentioned.
• Write down the ratio.
• Draw a cross joining the ratio to the two points
• Set up your answer by drawing a set of parentheses with two
vinculums separated by a comma
• Add the numbers in the ratio together to get the denominator
• Multiply along the cross and add to get the numerator
( 5,6) P = 3 × −3 + 1× 5 , 3 × 4 + 1× 6
( − 3,4)
4 4
− 4 18 9
= , = − 1,
1: 3 4 4 2
43. Type 2: External Division 2004 Extension 1 HSC Q1c)
Let A be the point ( 3,−1) and B be the point ( 9,2.) Find the coordinates
of the point P which divides AB externally in the ratio 5 : 2.
44. Type 2: External Division 2004 Extension 1 HSC Q1c)
Let A be the point ( 3,−1) and B be the point ( 9,2.) Find the coordinates
of the point P which divides AB externally in the ratio 5 : 2.
• Done exactly the same as internal division, except make one of
the numbers in the ratio negative
45. Type 2: External Division 2004 Extension 1 HSC Q1c)
Let A be the point ( 3,−1) and B be the point ( 9,2.) Find the coordinates
of the point P which divides AB externally in the ratio 5 : 2.
• Done exactly the same as internal division, except make one of
the numbers in the ratio negative
( 3,−1) ( 9,2)
46. Type 2: External Division 2004 Extension 1 HSC Q1c)
Let A be the point ( 3,−1) and B be the point ( 9,2.) Find the coordinates
of the point P which divides AB externally in the ratio 5 : 2.
• Done exactly the same as internal division, except make one of
the numbers in the ratio negative
( 3,−1) ( 9,2)
−5:2
47. Type 2: External Division 2004 Extension 1 HSC Q1c)
Let A be the point ( 3,−1) and B be the point ( 9,2.) Find the coordinates
of the point P which divides AB externally in the ratio 5 : 2.
• Done exactly the same as internal division, except make one of
the numbers in the ratio negative
( 3,−1) ( 9,2)
−5:2
48. Type 2: External Division 2004 Extension 1 HSC Q1c)
Let A be the point ( 3,−1) and B be the point ( 9,2.) Find the coordinates
of the point P which divides AB externally in the ratio 5 : 2.
• Done exactly the same as internal division, except make one of
the numbers in the ratio negative
( 3,−1) ( 9,2)
−5:2
P=
,
49. Type 2: External Division 2004 Extension 1 HSC Q1c)
Let A be the point ( 3,−1) and B be the point ( 9,2.) Find the coordinates
of the point P which divides AB externally in the ratio 5 : 2.
• Done exactly the same as internal division, except make one of
the numbers in the ratio negative
( 3,−1) ( 9,2)
−5:2
P=
,
−3 −3
50. Type 2: External Division 2004 Extension 1 HSC Q1c)
Let A be the point ( 3,−1) and B be the point ( 9,2.) Find the coordinates
of the point P which divides AB externally in the ratio 5 : 2.
• Done exactly the same as internal division, except make one of
the numbers in the ratio negative
( 3,−1) ( 9,2)
−5:2
2× 3 − 5× 9
P=
,
−3 −3
51. Type 2: External Division 2004 Extension 1 HSC Q1c)
Let A be the point ( 3,−1) and B be the point ( 9,2.) Find the coordinates
of the point P which divides AB externally in the ratio 5 : 2.
• Done exactly the same as internal division, except make one of
the numbers in the ratio negative
( 3,−1) ( 9,2)
−5:2
2 × 3 − 5 × 9 2 × −1 − 5 × 2
P=
,
−3 −3
52. Type 2: External Division 2004 Extension 1 HSC Q1c)
Let A be the point ( 3,−1) and B be the point ( 9,2.) Find the coordinates
of the point P which divides AB externally in the ratio 5 : 2.
• Done exactly the same as internal division, except make one of
the numbers in the ratio negative
( 3,−1) ( 9,2)
−5:2
2 × 3 − 5 × 9 2 × −1 − 5 × 2
P=
,
−3 −3
− 39 − 12
= ,
−3 −3
53. Type 2: External Division 2004 Extension 1 HSC Q1c)
Let A be the point ( 3,−1) and B be the point ( 9,2.) Find the coordinates
of the point P which divides AB externally in the ratio 5 : 2.
• Done exactly the same as internal division, except make one of
the numbers in the ratio negative
( 3,−1) ( 9,2)
−5:2
2 × 3 − 5 × 9 2 × −1 − 5 × 2
P=
,
−3 −3
− 39 − 12
= ,
−3 −3
= (13,4 )
54. Type 3: Find an endpoint of an interval 2005 Extension 1 HSC Q1e)
The point P(1,4) divides the line segment joining A( − 1,8)and B( x, y )
internally in the ratio 2 : 3.
Find the coordinates of the point B.
55. Type 3: Find an endpoint of an interval 2005 Extension 1 HSC Q1e)
The point P(1,4) divides the line segment joining A( − 1,8)and B( x, y )
internally in the ratio 2 : 3.
Find the coordinates of the point B.
• Draw the endpoints, ratio and cross the same as previously
56. Type 3: Find an endpoint of an interval 2005 Extension 1 HSC Q1e)
The point P(1,4) divides the line segment joining A( − 1,8)and B( x, y )
internally in the ratio 2 : 3.
Find the coordinates of the point B.
• Draw the endpoints, ratio and cross the same as previously
( x, y )
( − 1,8)
57. Type 3: Find an endpoint of an interval 2005 Extension 1 HSC Q1e)
The point P(1,4) divides the line segment joining A( − 1,8)and B( x, y )
internally in the ratio 2 : 3.
Find the coordinates of the point B.
• Draw the endpoints, ratio and cross the same as previously
( x, y )
( − 1,8)
2:3
58. Type 3: Find an endpoint of an interval 2005 Extension 1 HSC Q1e)
The point P(1,4) divides the line segment joining A( − 1,8)and B( x, y )
internally in the ratio 2 : 3.
Find the coordinates of the point B.
• Draw the endpoints, ratio and cross the same as previously
( x, y )
( − 1,8)
2:3
59. Type 3: Find an endpoint of an interval 2005 Extension 1 HSC Q1e)
The point P(1,4) divides the line segment joining A( − 1,8)and B( x, y )
internally in the ratio 2 : 3.
Find the coordinates of the point B.
• Draw the endpoints, ratio and cross the same as previously
• Create the fraction for the x value and equate it with the known value
( x, y )
( − 1,8)
2:3
60. Type 3: Find an endpoint of an interval 2005 Extension 1 HSC Q1e)
The point P(1,4) divides the line segment joining A( − 1,8)and B( x, y )
internally in the ratio 2 : 3.
Find the coordinates of the point B.
• Draw the endpoints, ratio and cross the same as previously
• Create the fraction for the x value and equate it with the known value
( x, y )
( − 1,8)
3 × −1 + 2 × x 2:3
1=
5
61. Type 3: Find an endpoint of an interval 2005 Extension 1 HSC Q1e)
The point P(1,4) divides the line segment joining A( − 1,8)and B( x, y )
internally in the ratio 2 : 3.
Find the coordinates of the point B.
• Draw the endpoints, ratio and cross the same as previously
• Create the fraction for the x value and equate it with the known value
( x, y )
( − 1,8)
3 × −1 + 2 × x 2:3
1=
5
5 = −3 + 2 x
62. Type 3: Find an endpoint of an interval 2005 Extension 1 HSC Q1e)
The point P(1,4) divides the line segment joining A( − 1,8)and B( x, y )
internally in the ratio 2 : 3.
Find the coordinates of the point B.
• Draw the endpoints, ratio and cross the same as previously
• Create the fraction for the x value and equate it with the known value
( x, y )
( − 1,8)
3 × −1 + 2 × x 2:3
1=
5
5 = −3 + 2 x
2x = 8
x=4
63. Type 3: Find an endpoint of an interval 2005 Extension 1 HSC Q1e)
The point P(1,4) divides the line segment joining A( − 1,8)and B( x, y )
internally in the ratio 2 : 3.
Find the coordinates of the point B.
• Draw the endpoints, ratio and cross the same as previously
• Create the fraction for the x value and equate it with the known value
• Repeat for the y value
( x, y )
( − 1,8)
3 × −1 + 2 × x 2:3
1=
5
5 = −3 + 2 x
2x = 8
x=4
64. Type 3: Find an endpoint of an interval 2005 Extension 1 HSC Q1e)
The point P(1,4) divides the line segment joining A( − 1,8)and B( x, y )
internally in the ratio 2 : 3.
Find the coordinates of the point B.
• Draw the endpoints, ratio and cross the same as previously
• Create the fraction for the x value and equate it with the known value
• Repeat for the y value
( x, y )
( − 1,8)
3 × −1 + 2 × x 3× 8 + 2 × y
2:3
1= 4=
5 5
5 = −3 + 2 x
2x = 8
x=4
65. Type 3: Find an endpoint of an interval 2005 Extension 1 HSC Q1e)
The point P(1,4) divides the line segment joining A( − 1,8)and B( x, y )
internally in the ratio 2 : 3.
Find the coordinates of the point B.
• Draw the endpoints, ratio and cross the same as previously
• Create the fraction for the x value and equate it with the known value
• Repeat for the y value
( x, y )
( − 1,8)
3 × −1 + 2 × x 3× 8 + 2 × y
2:3
1= 4=
5 5
5 = −3 + 2 x 20 = 24 + 2 y
2x = 8
x=4
66. Type 3: Find an endpoint of an interval 2005 Extension 1 HSC Q1e)
The point P(1,4) divides the line segment joining A( − 1,8)and B( x, y )
internally in the ratio 2 : 3.
Find the coordinates of the point B.
• Draw the endpoints, ratio and cross the same as previously
• Create the fraction for the x value and equate it with the known value
• Repeat for the y value
( x, y )
( − 1,8)
3 × −1 + 2 × x 3× 8 + 2 × y
2:3
1= 4=
5 5
5 = −3 + 2 x 20 = 24 + 2 y
2 y = −4
2x = 8
y = −2
x=4
67. Type 3: Find an endpoint of an interval 2005 Extension 1 HSC Q1e)
The point P(1,4) divides the line segment joining A( − 1,8)and B( x, y )
internally in the ratio 2 : 3.
Find the coordinates of the point B.
• Draw the endpoints, ratio and cross the same as previously
• Create the fraction for the x value and equate it with the known value
• Repeat for the y value
( x, y )
( − 1,8)
3 × −1 + 2 × x 3× 8 + 2 × y
2:3
1= 4=
5 5
5 = −3 + 2 x 20 = 24 + 2 y
2 y = −4
2x = 8
y = −2
x=4 ∴ B = ( 4,−2 )
68. Alternative
The point P(1,4) divides the line segment joining A( − 1,8)and B( x, y )
internally in the ratio 2 : 3.
Find the coordinates of the point B.
69. Alternative
The point P(1,4) divides the line segment joining A( − 1,8)and B( x, y )
internally in the ratio 2 : 3.
Find the coordinates of the point B.
A B
70. Alternative
The point P(1,4) divides the line segment joining A( − 1,8)and B( x, y )
internally in the ratio 2 : 3.
Find the coordinates of the point B.
If P divides AB internally in the ratio 2 : 3
A P B
2 3
71. Alternative
The point P(1,4) divides the line segment joining A( − 1,8)and B( x, y )
internally in the ratio 2 : 3.
Find the coordinates of the point B.
If P divides AB internally in the ratio 2 : 3
A P B
Then B divides AP externally in the ratio 5
:3
2 3
72. Alternative
The point P(1,4) divides the line segment joining A( − 1,8)and B( x, y )
internally in the ratio 2 : 3.
Find the coordinates of the point B.
If P divides AB internally in the ratio 2 : 3
A P B
Then B divides AP externally in the ratio 5
:3
2 3
( − 1,8) (1,4)
73. Alternative
The point P(1,4) divides the line segment joining A( − 1,8)and B( x, y )
internally in the ratio 2 : 3.
Find the coordinates of the point B.
If P divides AB internally in the ratio 2 : 3
A P B
Then B divides AP externally in the ratio 5
:3
2 3
( − 1,8) (1,4)
−5:3
74. Alternative
The point P(1,4) divides the line segment joining A( − 1,8)and B( x, y )
internally in the ratio 2 : 3.
Find the coordinates of the point B.
If P divides AB internally in the ratio 2 : 3
A P B
Then B divides AP externally in the ratio 5
:3
2 3
( − 1,8) (1,4)
−5:3
75. Alternative
The point P(1,4) divides the line segment joining A( − 1,8)and B( x, y )
internally in the ratio 2 : 3.
Find the coordinates of the point B.
If P divides AB internally in the ratio 2 : 3
A P B
Then B divides AP externally in the ratio 5
:3
2 3
( − 1,8) (1,4) B=
,
−5:3
76. Alternative
The point P(1,4) divides the line segment joining A( − 1,8)and B( x, y )
internally in the ratio 2 : 3.
Find the coordinates of the point B.
If P divides AB internally in the ratio 2 : 3
A P B
Then B divides AP externally in the ratio 5
:3
2 3
( − 1,8) (1,4) B=
,
−2 −2
−5:3
77. Alternative
The point P(1,4) divides the line segment joining A( − 1,8)and B( x, y )
internally in the ratio 2 : 3.
Find the coordinates of the point B.
If P divides AB internally in the ratio 2 : 3
A P B
Then B divides AP externally in the ratio 5
:3
2 3
B = 3 × −1 − 5 × 1 ,
( − 1,8) (1,4)
−2 −2
−5:3
78. Alternative
The point P(1,4) divides the line segment joining A( − 1,8)and B( x, y )
internally in the ratio 2 : 3.
Find the coordinates of the point B.
If P divides AB internally in the ratio 2 : 3
A P B
Then B divides AP externally in the ratio 5
:3
2 3
B = 3 × −1 − 5 × 1 , 3 × 8 − 5 × 4
( − 1,8) (1,4)
−2 −2
−5:3
79. Alternative
The point P(1,4) divides the line segment joining A( − 1,8)and B( x, y )
internally in the ratio 2 : 3.
Find the coordinates of the point B.
If P divides AB internally in the ratio 2 : 3
A P B
Then B divides AP externally in the ratio 5
:3
2 3
B = 3 × −1 − 5 × 1 , 3 × 8 − 5 × 4
( − 1,8) (1,4)
−2 −2
−8, 4
=
−5:3 − 2 − 2
80. Alternative
The point P(1,4) divides the line segment joining A( − 1,8)and B( x, y )
internally in the ratio 2 : 3.
Find the coordinates of the point B.
If P divides AB internally in the ratio 2 : 3
A P B
Then B divides AP externally in the ratio 5
:3
2 3
B = 3 × −1 − 5 × 1 , 3 × 8 − 5 × 4
( − 1,8) (1,4)
−2 −2
−8, 4
=
−5:3 − 2 − 2
= ( 4,−2 )
81. Type 4: Finding the ratio 1991 Extension 1 HSC Q1c)
The point P( − 3,8) divides the interval externally in the ratio k : 1.
If A is the point ( 6,−4 )and B is the point ( 0,4 ) find the value of k.
,
82. Type 4: Finding the ratio 1991 Extension 1 HSC Q1c)
The point P( − 3,8) divides the interval externally in the ratio k : 1.
If A is the point ( 6,−4 )and B is the point ( 0,4 ) find the value of k.
,
• Draw the endpoints, ratio and cross the same as usual
83. Type 4: Finding the ratio 1991 Extension 1 HSC Q1c)
The point P( − 3,8) divides the interval externally in the ratio k : 1.
If A is the point ( 6,−4 )and B is the point ( 0,4 ) find the value of k.
,
• Draw the endpoints, ratio and cross the same as usual
( 6,−4) ( 0,4)
84. Type 4: Finding the ratio 1991 Extension 1 HSC Q1c)
The point P( − 3,8) divides the interval externally in the ratio k : 1.
If A is the point ( 6,−4 )and B is the point ( 0,4 ) find the value of k.
,
• Draw the endpoints, ratio and cross the same as usual
( 6,−4) ( 0,4)
− k :1
85. Type 4: Finding the ratio 1991 Extension 1 HSC Q1c)
The point P( − 3,8) divides the interval externally in the ratio k : 1.
If A is the point ( 6,−4 )and B is the point ( 0,4 ) find the value of k.
,
• Draw the endpoints, ratio and cross the same as usual
( 6,−4) ( 0,4)
− k :1
86. Type 4: Finding the ratio 1991 Extension 1 HSC Q1c)
The point P( − 3,8) divides the interval externally in the ratio k : 1.
If A is the point ( 6,−4 )and B is the point ( 0,4 ) find the value of k.
,
• Draw the endpoints, ratio and cross the same as usual
• Create the fraction for the either the x value or the y value (it does not
matter which one) and equate it with the known value
( 6,−4) ( 0,4)
− k :1
87. Type 4: Finding the ratio 1991 Extension 1 HSC Q1c)
The point P( − 3,8) divides the interval externally in the ratio k : 1.
If A is the point ( 6,−4 )and B is the point ( 0,4 ) find the value of k.
,
• Draw the endpoints, ratio and cross the same as usual
• Create the fraction for the either the x value or the y value (it does not
matter which one) and equate it with the known value
( 6,−4) ( 0,4)
− k :1
1× 6 + − k × 0
−3 =
− k +1
88. Type 4: Finding the ratio 1991 Extension 1 HSC Q1c)
The point P( − 3,8) divides the interval externally in the ratio k : 1.
If A is the point ( 6,−4 )and B is the point ( 0,4 ) find the value of k.
,
• Draw the endpoints, ratio and cross the same as usual
• Create the fraction for the either the x value or the y value (it does not
matter which one) and equate it with the known value
( 6,−4) ( 0,4)
− k :1
1× 6 + − k × 0
−3 =
− k +1
3k − 3 = 6
89. Type 4: Finding the ratio 1991 Extension 1 HSC Q1c)
The point P( − 3,8) divides the interval externally in the ratio k : 1.
If A is the point ( 6,−4 )and B is the point ( 0,4 ) find the value of k.
,
• Draw the endpoints, ratio and cross the same as usual
• Create the fraction for the either the x value or the y value (it does not
matter which one) and equate it with the known value
( 6,−4) ( 0,4)
− k :1
1× 6 + − k × 0
−3 =
− k +1
3k − 3 = 6
3k = 9
k =3
90. Type 4: Finding the ratio 1991 Extension 1 HSC Q1c)
The point P( − 3,8) divides the interval externally in the ratio k : 1.
If A is the point ( 6,−4 )and B is the point ( 0,4 ) find the value of k.
,
• Draw the endpoints, ratio and cross the same as usual
• Create the fraction for the either the x value or the y value (it does not
matter which one) and equate it with the known value
( 6,−4) ( 0,4)
− k :1
1× 6 + − k × 0
−3 =
− k +1
3k − 3 = 6
3k = 9
∴ P divides AB externally in the ratio 3 : 1
k =3
91. Type 4: Finding the ratio 1991 Extension 1 HSC Q1c)
The point P( − 3,8) divides the interval externally in the ratio k : 1.
If A is the point ( 6,−4 )and B is the point ( 0,4 ) find the value of k.
,
• Draw the endpoints, ratio and cross the same as usual
• Create the fraction for the either the x value or the y value (it does not
matter which one) and equate it with the known value
( 6,−4) ( 0,4)
Exercise 5A;
1ad, 2ad,
− k :1 3 i, iii in all, 4ace,
1× 6 + − k × 0
−3 = 5 i, ii ace, 6bd,
− k +1 8, 9, 11, 13b, 16,
3k − 3 = 6 17, 20, 21, 23, 24
3k = 9
∴ P divides AB externally in the ratio 3 : 1
k =3
