11x1 t05 01 division of an interval (2011)
TRANSCRIPT
![Page 1: 11X1 T05 01 division of an interval (2011)](https://reader036.vdocument.in/reader036/viewer/2022062405/5559cbedd8b42a93208b4890/html5/thumbnails/1.jpg)
Coordinate Geometry
![Page 2: 11X1 T05 01 division of an interval (2011)](https://reader036.vdocument.in/reader036/viewer/2022062405/5559cbedd8b42a93208b4890/html5/thumbnails/2.jpg)
Coordinate GeometryDistance Formula
![Page 3: 11X1 T05 01 division of an interval (2011)](https://reader036.vdocument.in/reader036/viewer/2022062405/5559cbedd8b42a93208b4890/html5/thumbnails/3.jpg)
Coordinate GeometryDistance Formula
2 22 1 2 1d x x y y
![Page 4: 11X1 T05 01 division of an interval (2011)](https://reader036.vdocument.in/reader036/viewer/2022062405/5559cbedd8b42a93208b4890/html5/thumbnails/4.jpg)
Coordinate GeometryDistance Formula
2 22 1 2 1d x x y y
e.g. Find the distance between (–1,3) and (3,5)
![Page 5: 11X1 T05 01 division of an interval (2011)](https://reader036.vdocument.in/reader036/viewer/2022062405/5559cbedd8b42a93208b4890/html5/thumbnails/5.jpg)
Coordinate GeometryDistance Formula
2 22 1 2 1d x x y y
e.g. Find the distance between (–1,3) and (3,5)
2 25 3 3 1d
![Page 6: 11X1 T05 01 division of an interval (2011)](https://reader036.vdocument.in/reader036/viewer/2022062405/5559cbedd8b42a93208b4890/html5/thumbnails/6.jpg)
Coordinate GeometryDistance Formula
2 22 1 2 1d x x y y
e.g. Find the distance between (–1,3) and (3,5)
2 25 3 3 1d
2 22 4
20
2 5 units
![Page 7: 11X1 T05 01 division of an interval (2011)](https://reader036.vdocument.in/reader036/viewer/2022062405/5559cbedd8b42a93208b4890/html5/thumbnails/7.jpg)
Coordinate GeometryDistance Formula
2 22 1 2 1d x x y y
e.g. Find the distance between (–1,3) and (3,5)
2 25 3 3 1d
2 22 4
20
2 5 units
The distance formula is finding the length of the hypotenuse,
using Pythagoras
![Page 8: 11X1 T05 01 division of an interval (2011)](https://reader036.vdocument.in/reader036/viewer/2022062405/5559cbedd8b42a93208b4890/html5/thumbnails/8.jpg)
Coordinate GeometryDistance Formula
2 22 1 2 1d x x y y
e.g. Find the distance between (–1,3) and (3,5)
2 25 3 3 1d
2 22 4
20
2 5 units
Midpoint Formula
The distance formula is finding the length of the hypotenuse,
using Pythagoras
![Page 9: 11X1 T05 01 division of an interval (2011)](https://reader036.vdocument.in/reader036/viewer/2022062405/5559cbedd8b42a93208b4890/html5/thumbnails/9.jpg)
Coordinate GeometryDistance Formula
2 22 1 2 1d x x y y
e.g. Find the distance between (–1,3) and (3,5)
2 25 3 3 1d
2 22 4
20
2 5 units
Midpoint Formula
1 2 1 2,2 2
x x y yM
The distance formula is finding the length of the hypotenuse,
using Pythagoras
![Page 10: 11X1 T05 01 division of an interval (2011)](https://reader036.vdocument.in/reader036/viewer/2022062405/5559cbedd8b42a93208b4890/html5/thumbnails/10.jpg)
Coordinate GeometryDistance Formula
2 22 1 2 1d x x y y
e.g. Find the distance between (–1,3) and (3,5)
2 25 3 3 1d
2 22 4
20
2 5 units
Midpoint Formula
1 2 1 2,2 2
x x y yM
e.g. Find the midpoint of (3,4) and (–2 ,1)
The distance formula is finding the length of the hypotenuse,
using Pythagoras
![Page 11: 11X1 T05 01 division of an interval (2011)](https://reader036.vdocument.in/reader036/viewer/2022062405/5559cbedd8b42a93208b4890/html5/thumbnails/11.jpg)
Coordinate GeometryDistance Formula
2 22 1 2 1d x x y y
e.g. Find the distance between (–1,3) and (3,5)
2 25 3 3 1d
2 22 4
20
2 5 units
Midpoint Formula
1 2 1 2,2 2
x x y yM
e.g. Find the midpoint of (3,4) and (–2 ,1)
3 2 4 1,2 2
M
The distance formula is finding the length of the hypotenuse,
using Pythagoras
![Page 12: 11X1 T05 01 division of an interval (2011)](https://reader036.vdocument.in/reader036/viewer/2022062405/5559cbedd8b42a93208b4890/html5/thumbnails/12.jpg)
Coordinate GeometryDistance Formula
2 22 1 2 1d x x y y
e.g. Find the distance between (–1,3) and (3,5)
2 25 3 3 1d
2 22 4
20
2 5 units
Midpoint Formula
1 2 1 2,2 2
x x y yM
e.g. Find the midpoint of (3,4) and (–2 ,1)
3 2 4 1,2 2
M
1 5,2 2
The distance formula is finding the length of the hypotenuse,
using Pythagoras
![Page 13: 11X1 T05 01 division of an interval (2011)](https://reader036.vdocument.in/reader036/viewer/2022062405/5559cbedd8b42a93208b4890/html5/thumbnails/13.jpg)
Coordinate GeometryDistance Formula
2 22 1 2 1d x x y y
e.g. Find the distance between (–1,3) and (3,5)
2 25 3 3 1d
2 22 4
20
2 5 units
Midpoint Formula
1 2 1 2,2 2
x x y yM
e.g. Find the midpoint of (3,4) and (–2 ,1)
3 2 4 1,2 2
M
1 5,2 2
The distance formula is finding the length of the hypotenuse,
using Pythagoras
The midpoint formula averages the x and y values
![Page 14: 11X1 T05 01 division of an interval (2011)](https://reader036.vdocument.in/reader036/viewer/2022062405/5559cbedd8b42a93208b4890/html5/thumbnails/14.jpg)
Division Of An Interval
![Page 15: 11X1 T05 01 division of an interval (2011)](https://reader036.vdocument.in/reader036/viewer/2022062405/5559cbedd8b42a93208b4890/html5/thumbnails/15.jpg)
Division Of An IntervalMathematics (2 unit) division of an interval questions are restricted to midpoint questions i.e. dividing in the ratio 1:1
![Page 16: 11X1 T05 01 division of an interval (2011)](https://reader036.vdocument.in/reader036/viewer/2022062405/5559cbedd8b42a93208b4890/html5/thumbnails/16.jpg)
Division Of An IntervalMathematics (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.
![Page 17: 11X1 T05 01 division of an interval (2011)](https://reader036.vdocument.in/reader036/viewer/2022062405/5559cbedd8b42a93208b4890/html5/thumbnails/17.jpg)
Division Of An IntervalMathematics (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
![Page 18: 11X1 T05 01 division of an interval (2011)](https://reader036.vdocument.in/reader036/viewer/2022062405/5559cbedd8b42a93208b4890/html5/thumbnails/18.jpg)
Division Of An IntervalMathematics (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 YA
![Page 19: 11X1 T05 01 division of an interval (2011)](https://reader036.vdocument.in/reader036/viewer/2022062405/5559cbedd8b42a93208b4890/html5/thumbnails/19.jpg)
Division Of An IntervalMathematics (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 YA
A divides XY internally in the ratio m:n
![Page 20: 11X1 T05 01 division of an interval (2011)](https://reader036.vdocument.in/reader036/viewer/2022062405/5559cbedd8b42a93208b4890/html5/thumbnails/20.jpg)
Division Of An IntervalMathematics (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 YA
A divides XY internally in the ratio m:n
m
n
![Page 21: 11X1 T05 01 division of an interval (2011)](https://reader036.vdocument.in/reader036/viewer/2022062405/5559cbedd8b42a93208b4890/html5/thumbnails/21.jpg)
Division Of An IntervalMathematics (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 YA
A divides XY internally in the ratio m:n
m
n
OR
![Page 22: 11X1 T05 01 division of an interval (2011)](https://reader036.vdocument.in/reader036/viewer/2022062405/5559cbedd8b42a93208b4890/html5/thumbnails/22.jpg)
Division Of An IntervalMathematics (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 YA
A divides XY internally in the ratio m:n
m
n
ORA divides YX internally in the ratio n:m
![Page 23: 11X1 T05 01 division of an interval (2011)](https://reader036.vdocument.in/reader036/viewer/2022062405/5559cbedd8b42a93208b4890/html5/thumbnails/23.jpg)
Division Of An IntervalMathematics (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 YA
X Y
A divides XY internally in the ratio m:n
m
n
ORA divides YX internally in the ratio n:m
![Page 24: 11X1 T05 01 division of an interval (2011)](https://reader036.vdocument.in/reader036/viewer/2022062405/5559cbedd8b42a93208b4890/html5/thumbnails/24.jpg)
Division Of An IntervalMathematics (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 YA
X Y A
A divides XY internally in the ratio m:n
m
n
ORA divides YX internally in the ratio n:m
![Page 25: 11X1 T05 01 division of an interval (2011)](https://reader036.vdocument.in/reader036/viewer/2022062405/5559cbedd8b42a93208b4890/html5/thumbnails/25.jpg)
Division Of An IntervalMathematics (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 YA
X Y A
A divides XY internally in the ratio m:n
m
n
ORA divides YX internally in the ratio n:m
A divides XY externally in the ratio m:n
![Page 26: 11X1 T05 01 division of an interval (2011)](https://reader036.vdocument.in/reader036/viewer/2022062405/5559cbedd8b42a93208b4890/html5/thumbnails/26.jpg)
Division Of An IntervalMathematics (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 YA
X Y A
A divides XY internally in the ratio m:n
m
n
ORA divides YX internally in the ratio n:m
A divides XY externally in the ratio m:n
m
n
![Page 27: 11X1 T05 01 division of an interval (2011)](https://reader036.vdocument.in/reader036/viewer/2022062405/5559cbedd8b42a93208b4890/html5/thumbnails/27.jpg)
Type 1: Internal Division
Find the coordinates of P that divides the interval joining and internally in the ratio 1 : 3
4,3
2003 Extension 1 HSC Q1c)
6,5
![Page 28: 11X1 T05 01 division of an interval (2011)](https://reader036.vdocument.in/reader036/viewer/2022062405/5559cbedd8b42a93208b4890/html5/thumbnails/28.jpg)
Type 1: Internal Division
Find the coordinates of P that divides the interval joining and internally in the ratio 1 : 3
4,3
2003 Extension 1 HSC Q1c)
6,5• Write down the endpoints of the interval in the same order as
they are mentioned.
![Page 29: 11X1 T05 01 division of an interval (2011)](https://reader036.vdocument.in/reader036/viewer/2022062405/5559cbedd8b42a93208b4890/html5/thumbnails/29.jpg)
Type 1: Internal Division
Find the coordinates of P that divides the interval joining and internally in the ratio 1 : 3
4,3
2003 Extension 1 HSC Q1c)
6,5• Write down the endpoints of the interval in the same order as
they are mentioned.
4,3 6,5
![Page 30: 11X1 T05 01 division of an interval (2011)](https://reader036.vdocument.in/reader036/viewer/2022062405/5559cbedd8b42a93208b4890/html5/thumbnails/30.jpg)
Type 1: Internal Division
Find the coordinates of P that divides the interval joining and internally in the ratio 1 : 3
4,3
2003 Extension 1 HSC Q1c)
6,5• Write down the endpoints of the interval in the same order as
they are mentioned.
4,3 6,5
• Write down the ratio.
![Page 31: 11X1 T05 01 division of an interval (2011)](https://reader036.vdocument.in/reader036/viewer/2022062405/5559cbedd8b42a93208b4890/html5/thumbnails/31.jpg)
Type 1: Internal Division
Find the coordinates of P that divides the interval joining and internally in the ratio 1 : 3
4,3
2003 Extension 1 HSC Q1c)
6,5• Write down the endpoints of the interval in the same order as
they are mentioned.
4,3 6,5
• Write down the ratio.
3:1
![Page 32: 11X1 T05 01 division of an interval (2011)](https://reader036.vdocument.in/reader036/viewer/2022062405/5559cbedd8b42a93208b4890/html5/thumbnails/32.jpg)
Type 1: Internal Division
Find the coordinates of P that divides the interval joining and internally in the ratio 1 : 3
4,3
2003 Extension 1 HSC Q1c)
6,5• Write down the endpoints of the interval in the same order as
they are mentioned.
4,3 6,5
• Write down the ratio.
3:1
• Draw a cross joining the ratio to the two points
![Page 33: 11X1 T05 01 division of an interval (2011)](https://reader036.vdocument.in/reader036/viewer/2022062405/5559cbedd8b42a93208b4890/html5/thumbnails/33.jpg)
Type 1: Internal Division
Find the coordinates of P that divides the interval joining and internally in the ratio 1 : 3
4,3
2003 Extension 1 HSC Q1c)
6,5• Write down the endpoints of the interval in the same order as
they are mentioned.
4,3 6,5
• Write down the ratio.
3:1
• Draw a cross joining the ratio to the two points
![Page 34: 11X1 T05 01 division of an interval (2011)](https://reader036.vdocument.in/reader036/viewer/2022062405/5559cbedd8b42a93208b4890/html5/thumbnails/34.jpg)
Type 1: Internal Division
Find the coordinates of P that divides the interval joining and internally in the ratio 1 : 3
4,3
2003 Extension 1 HSC Q1c)
6,5• Write down the endpoints of the interval in the same order as
they are mentioned.
4,3 6,5
• Write down the ratio.
3:1
• 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
![Page 35: 11X1 T05 01 division of an interval (2011)](https://reader036.vdocument.in/reader036/viewer/2022062405/5559cbedd8b42a93208b4890/html5/thumbnails/35.jpg)
Type 1: Internal Division
Find the coordinates of P that divides the interval joining and internally in the ratio 1 : 3
4,3
2003 Extension 1 HSC Q1c)
6,5• Write down the endpoints of the interval in the same order as
they are mentioned.
4,3 6,5
• Write down the ratio.
3:1
• 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
, P
![Page 36: 11X1 T05 01 division of an interval (2011)](https://reader036.vdocument.in/reader036/viewer/2022062405/5559cbedd8b42a93208b4890/html5/thumbnails/36.jpg)
Type 1: Internal Division
Find the coordinates of P that divides the interval joining and internally in the ratio 1 : 3
4,3
2003 Extension 1 HSC Q1c)
6,5• Write down the endpoints of the interval in the same order as
they are mentioned.
4,3 6,5
• Write down the ratio.
3:1
• 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
, P
• Add the numbers in the ratio together to get the denominator
![Page 37: 11X1 T05 01 division of an interval (2011)](https://reader036.vdocument.in/reader036/viewer/2022062405/5559cbedd8b42a93208b4890/html5/thumbnails/37.jpg)
Type 1: Internal Division
Find the coordinates of P that divides the interval joining and internally in the ratio 1 : 3
4,3
2003 Extension 1 HSC Q1c)
6,5• Write down the endpoints of the interval in the same order as
they are mentioned.
4,3 6,5
• Write down the ratio.
3:1
• 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
, P
• Add the numbers in the ratio together to get the denominator
4 4
![Page 38: 11X1 T05 01 division of an interval (2011)](https://reader036.vdocument.in/reader036/viewer/2022062405/5559cbedd8b42a93208b4890/html5/thumbnails/38.jpg)
Type 1: Internal Division
Find the coordinates of P that divides the interval joining and internally in the ratio 1 : 3
4,3
2003 Extension 1 HSC Q1c)
6,5• Write down the endpoints of the interval in the same order as
they are mentioned.
4,3 6,5
• Write down the ratio.
3:1
• 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
, P
• Add the numbers in the ratio together to get the denominator
4 4
• Multiply along the cross and add to get the numerator
![Page 39: 11X1 T05 01 division of an interval (2011)](https://reader036.vdocument.in/reader036/viewer/2022062405/5559cbedd8b42a93208b4890/html5/thumbnails/39.jpg)
Type 1: Internal Division
Find the coordinates of P that divides the interval joining and internally in the ratio 1 : 3
4,3
2003 Extension 1 HSC Q1c)
6,5• Write down the endpoints of the interval in the same order as
they are mentioned.
4,3 6,5
• Write down the ratio.
3:1
• 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
, P
• Add the numbers in the ratio together to get the denominator
4 4
• Multiply along the cross and add to get the numerator5133
![Page 40: 11X1 T05 01 division of an interval (2011)](https://reader036.vdocument.in/reader036/viewer/2022062405/5559cbedd8b42a93208b4890/html5/thumbnails/40.jpg)
Type 1: Internal Division
Find the coordinates of P that divides the interval joining and internally in the ratio 1 : 3
4,3
2003 Extension 1 HSC Q1c)
6,5• Write down the endpoints of the interval in the same order as
they are mentioned.
4,3 6,5
• Write down the ratio.
3:1
• 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
, P
• Add the numbers in the ratio together to get the denominator
4 4
• Multiply along the cross and add to get the numerator5133 6143
![Page 41: 11X1 T05 01 division of an interval (2011)](https://reader036.vdocument.in/reader036/viewer/2022062405/5559cbedd8b42a93208b4890/html5/thumbnails/41.jpg)
Type 1: Internal Division
Find the coordinates of P that divides the interval joining and internally in the ratio 1 : 3
4,3
2003 Extension 1 HSC Q1c)
6,5• Write down the endpoints of the interval in the same order as
they are mentioned.
4,3 6,5
• Write down the ratio.
3:1
• 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
, P
• Add the numbers in the ratio together to get the denominator
4 4
• Multiply along the cross and add to get the numerator5133 6143
418,
44
![Page 42: 11X1 T05 01 division of an interval (2011)](https://reader036.vdocument.in/reader036/viewer/2022062405/5559cbedd8b42a93208b4890/html5/thumbnails/42.jpg)
Type 1: Internal Division
Find the coordinates of P that divides the interval joining and internally in the ratio 1 : 3
4,3
2003 Extension 1 HSC Q1c)
6,5• Write down the endpoints of the interval in the same order as
they are mentioned.
4,3 6,5
• Write down the ratio.
3:1
• 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
, P
• Add the numbers in the ratio together to get the denominator
4 4
• Multiply along the cross and add to get the numerator5133 6143
418,
44
29,1
![Page 43: 11X1 T05 01 division of an interval (2011)](https://reader036.vdocument.in/reader036/viewer/2022062405/5559cbedd8b42a93208b4890/html5/thumbnails/43.jpg)
Type 2: External Division
Let A be the point and B be the point . Find the coordinates of the point P which divides AB externally in the ratio 5 : 2.
1,3
2004 Extension 1 HSC Q1c)
2,9
![Page 44: 11X1 T05 01 division of an interval (2011)](https://reader036.vdocument.in/reader036/viewer/2022062405/5559cbedd8b42a93208b4890/html5/thumbnails/44.jpg)
Type 2: External Division
Let A be the point and B be the point . Find the coordinates of the point P which divides AB externally in the ratio 5 : 2.
1,3
2004 Extension 1 HSC Q1c)
2,9
• Done exactly the same as internal division, except make one of the numbers in the ratio negative
![Page 45: 11X1 T05 01 division of an interval (2011)](https://reader036.vdocument.in/reader036/viewer/2022062405/5559cbedd8b42a93208b4890/html5/thumbnails/45.jpg)
Type 2: External Division
Let A be the point and B be the point . Find the coordinates of the point P which divides AB externally in the ratio 5 : 2.
1,3
2004 Extension 1 HSC Q1c)
2,9
• Done exactly the same as internal division, except make one of the numbers in the ratio negative
1,3 2,9
![Page 46: 11X1 T05 01 division of an interval (2011)](https://reader036.vdocument.in/reader036/viewer/2022062405/5559cbedd8b42a93208b4890/html5/thumbnails/46.jpg)
Type 2: External Division
Let A be the point and B be the point . Find the coordinates of the point P which divides AB externally in the ratio 5 : 2.
1,3
2004 Extension 1 HSC Q1c)
2,9
• Done exactly the same as internal division, except make one of the numbers in the ratio negative
1,3 2,9
2:5
![Page 47: 11X1 T05 01 division of an interval (2011)](https://reader036.vdocument.in/reader036/viewer/2022062405/5559cbedd8b42a93208b4890/html5/thumbnails/47.jpg)
Type 2: External Division
Let A be the point and B be the point . Find the coordinates of the point P which divides AB externally in the ratio 5 : 2.
1,3
2004 Extension 1 HSC Q1c)
2,9
• Done exactly the same as internal division, except make one of the numbers in the ratio negative
1,3 2,9
2:5
![Page 48: 11X1 T05 01 division of an interval (2011)](https://reader036.vdocument.in/reader036/viewer/2022062405/5559cbedd8b42a93208b4890/html5/thumbnails/48.jpg)
Type 2: External Division
Let A be the point and B be the point . Find the coordinates of the point P which divides AB externally in the ratio 5 : 2.
1,3
2004 Extension 1 HSC Q1c)
2,9
• Done exactly the same as internal division, except make one of the numbers in the ratio negative
1,3 2,9
2:5
, P
![Page 49: 11X1 T05 01 division of an interval (2011)](https://reader036.vdocument.in/reader036/viewer/2022062405/5559cbedd8b42a93208b4890/html5/thumbnails/49.jpg)
Type 2: External Division
Let A be the point and B be the point . Find the coordinates of the point P which divides AB externally in the ratio 5 : 2.
1,3
2004 Extension 1 HSC Q1c)
2,9
• Done exactly the same as internal division, except make one of the numbers in the ratio negative
1,3 2,9
2:5
, P 3 3
![Page 50: 11X1 T05 01 division of an interval (2011)](https://reader036.vdocument.in/reader036/viewer/2022062405/5559cbedd8b42a93208b4890/html5/thumbnails/50.jpg)
Type 2: External Division
Let A be the point and B be the point . Find the coordinates of the point P which divides AB externally in the ratio 5 : 2.
1,3
2004 Extension 1 HSC Q1c)
2,9
• Done exactly the same as internal division, except make one of the numbers in the ratio negative
1,3 2,9
2:5
, P 3 39532
![Page 51: 11X1 T05 01 division of an interval (2011)](https://reader036.vdocument.in/reader036/viewer/2022062405/5559cbedd8b42a93208b4890/html5/thumbnails/51.jpg)
Type 2: External Division
Let A be the point and B be the point . Find the coordinates of the point P which divides AB externally in the ratio 5 : 2.
1,3
2004 Extension 1 HSC Q1c)
2,9
• Done exactly the same as internal division, except make one of the numbers in the ratio negative
1,3 2,9
2:5
, P 3 39532 2512
![Page 52: 11X1 T05 01 division of an interval (2011)](https://reader036.vdocument.in/reader036/viewer/2022062405/5559cbedd8b42a93208b4890/html5/thumbnails/52.jpg)
Type 2: External Division
Let A be the point and B be the point . Find the coordinates of the point P which divides AB externally in the ratio 5 : 2.
1,3
2004 Extension 1 HSC Q1c)
2,9
• Done exactly the same as internal division, except make one of the numbers in the ratio negative
1,3 2,9
2:5
, P 3 39532 2512
3
12,3
39
![Page 53: 11X1 T05 01 division of an interval (2011)](https://reader036.vdocument.in/reader036/viewer/2022062405/5559cbedd8b42a93208b4890/html5/thumbnails/53.jpg)
Type 2: External Division
Let A be the point and B be the point . Find the coordinates of the point P which divides AB externally in the ratio 5 : 2.
1,3
2004 Extension 1 HSC Q1c)
2,9
• Done exactly the same as internal division, except make one of the numbers in the ratio negative
1,3 2,9
2:5
, P 3 39532 2512
3
12,3
39
4,13
![Page 54: 11X1 T05 01 division of an interval (2011)](https://reader036.vdocument.in/reader036/viewer/2022062405/5559cbedd8b42a93208b4890/html5/thumbnails/54.jpg)
Type 3: Find an endpoint of an interval
The point divides the line segment joining and internally in the ratio 2 : 3.
Find the coordinates of the point B.
4,1P
2005 Extension 1 HSC Q1e)
8,1A yxB ,
![Page 55: 11X1 T05 01 division of an interval (2011)](https://reader036.vdocument.in/reader036/viewer/2022062405/5559cbedd8b42a93208b4890/html5/thumbnails/55.jpg)
Type 3: Find an endpoint of an interval
The point divides the line segment joining and internally in the ratio 2 : 3.
Find the coordinates of the point B.
4,1P
2005 Extension 1 HSC Q1e)
8,1A
• Draw the endpoints, ratio and cross the same as previously
yxB ,
![Page 56: 11X1 T05 01 division of an interval (2011)](https://reader036.vdocument.in/reader036/viewer/2022062405/5559cbedd8b42a93208b4890/html5/thumbnails/56.jpg)
Type 3: Find an endpoint of an interval
The point divides the line segment joining and internally in the ratio 2 : 3.
Find the coordinates of the point B.
4,1P
2005 Extension 1 HSC Q1e)
8,1A
• Draw the endpoints, ratio and cross the same as previously
8,1 yx,
yxB ,
![Page 57: 11X1 T05 01 division of an interval (2011)](https://reader036.vdocument.in/reader036/viewer/2022062405/5559cbedd8b42a93208b4890/html5/thumbnails/57.jpg)
Type 3: Find an endpoint of an interval
The point divides the line segment joining and internally in the ratio 2 : 3.
Find the coordinates of the point B.
4,1P
2005 Extension 1 HSC Q1e)
8,1A
• Draw the endpoints, ratio and cross the same as previously
8,1 yx,
3:2
yxB ,
![Page 58: 11X1 T05 01 division of an interval (2011)](https://reader036.vdocument.in/reader036/viewer/2022062405/5559cbedd8b42a93208b4890/html5/thumbnails/58.jpg)
Type 3: Find an endpoint of an interval
The point divides the line segment joining and internally in the ratio 2 : 3.
Find the coordinates of the point B.
4,1P
2005 Extension 1 HSC Q1e)
8,1A
• Draw the endpoints, ratio and cross the same as previously
8,1 yx,
3:2
yxB ,
![Page 59: 11X1 T05 01 division of an interval (2011)](https://reader036.vdocument.in/reader036/viewer/2022062405/5559cbedd8b42a93208b4890/html5/thumbnails/59.jpg)
Type 3: Find an endpoint of an interval
The point divides the line segment joining and internally in the ratio 2 : 3.
Find the coordinates of the point B.
4,1P
2005 Extension 1 HSC Q1e)
8,1A
• Draw the endpoints, ratio and cross the same as previously
8,1 yx,
3:2
yxB ,
• Create the fraction for the x value and equate it with the known value
![Page 60: 11X1 T05 01 division of an interval (2011)](https://reader036.vdocument.in/reader036/viewer/2022062405/5559cbedd8b42a93208b4890/html5/thumbnails/60.jpg)
Type 3: Find an endpoint of an interval
The point divides the line segment joining and internally in the ratio 2 : 3.
Find the coordinates of the point B.
4,1P
2005 Extension 1 HSC Q1e)
8,1A
• Draw the endpoints, ratio and cross the same as previously
8,1 yx,
3:25
2131 x
yxB ,
• Create the fraction for the x value and equate it with the known value
![Page 61: 11X1 T05 01 division of an interval (2011)](https://reader036.vdocument.in/reader036/viewer/2022062405/5559cbedd8b42a93208b4890/html5/thumbnails/61.jpg)
Type 3: Find an endpoint of an interval
The point divides the line segment joining and internally in the ratio 2 : 3.
Find the coordinates of the point B.
4,1P
2005 Extension 1 HSC Q1e)
8,1A
• Draw the endpoints, ratio and cross the same as previously
8,1 yx,
3:25
2131 x
yxB ,
• Create the fraction for the x value and equate it with the known value
x235
![Page 62: 11X1 T05 01 division of an interval (2011)](https://reader036.vdocument.in/reader036/viewer/2022062405/5559cbedd8b42a93208b4890/html5/thumbnails/62.jpg)
Type 3: Find an endpoint of an interval
The point divides the line segment joining and internally in the ratio 2 : 3.
Find the coordinates of the point B.
4,1P
2005 Extension 1 HSC Q1e)
8,1A
• Draw the endpoints, ratio and cross the same as previously
8,1 yx,
3:25
2131 x
yxB ,
• Create the fraction for the x value and equate it with the known value
x235
482
xx
![Page 63: 11X1 T05 01 division of an interval (2011)](https://reader036.vdocument.in/reader036/viewer/2022062405/5559cbedd8b42a93208b4890/html5/thumbnails/63.jpg)
Type 3: Find an endpoint of an interval
The point divides the line segment joining and internally in the ratio 2 : 3.
Find the coordinates of the point B.
4,1P
2005 Extension 1 HSC Q1e)
8,1A
• Draw the endpoints, ratio and cross the same as previously
8,1 yx,
3:25
2131 x
yxB ,
• Create the fraction for the x value and equate it with the known value• Repeat for the y value
x235
482
xx
![Page 64: 11X1 T05 01 division of an interval (2011)](https://reader036.vdocument.in/reader036/viewer/2022062405/5559cbedd8b42a93208b4890/html5/thumbnails/64.jpg)
Type 3: Find an endpoint of an interval
The point divides the line segment joining and internally in the ratio 2 : 3.
Find the coordinates of the point B.
4,1P
2005 Extension 1 HSC Q1e)
8,1A
• Draw the endpoints, ratio and cross the same as previously
8,1 yx,
3:25
2131 x
52834 y
yxB ,
• Create the fraction for the x value and equate it with the known value• Repeat for the y value
x235
482
xx
![Page 65: 11X1 T05 01 division of an interval (2011)](https://reader036.vdocument.in/reader036/viewer/2022062405/5559cbedd8b42a93208b4890/html5/thumbnails/65.jpg)
Type 3: Find an endpoint of an interval
The point divides the line segment joining and internally in the ratio 2 : 3.
Find the coordinates of the point B.
4,1P
2005 Extension 1 HSC Q1e)
8,1A
• Draw the endpoints, ratio and cross the same as previously
8,1 yx,
3:25
2131 x
52834 y
yxB ,
• Create the fraction for the x value and equate it with the known value• Repeat for the y value
x235
482
xx
y22420
![Page 66: 11X1 T05 01 division of an interval (2011)](https://reader036.vdocument.in/reader036/viewer/2022062405/5559cbedd8b42a93208b4890/html5/thumbnails/66.jpg)
Type 3: Find an endpoint of an interval
The point divides the line segment joining and internally in the ratio 2 : 3.
Find the coordinates of the point B.
4,1P
2005 Extension 1 HSC Q1e)
8,1A
• Draw the endpoints, ratio and cross the same as previously
8,1 yx,
3:25
2131 x
52834 y
yxB ,
• Create the fraction for the x value and equate it with the known value• Repeat for the y value
x235
482
xx
y22420
242
yy
![Page 67: 11X1 T05 01 division of an interval (2011)](https://reader036.vdocument.in/reader036/viewer/2022062405/5559cbedd8b42a93208b4890/html5/thumbnails/67.jpg)
Type 3: Find an endpoint of an interval
The point divides the line segment joining and internally in the ratio 2 : 3.
Find the coordinates of the point B.
4,1P
2005 Extension 1 HSC Q1e)
8,1A
• Draw the endpoints, ratio and cross the same as previously
8,1 yx,
3:25
2131 x
52834 y
2,4 B
yxB ,
• Create the fraction for the x value and equate it with the known value• Repeat for the y value
x235
482
xx
y22420
242
yy
![Page 68: 11X1 T05 01 division of an interval (2011)](https://reader036.vdocument.in/reader036/viewer/2022062405/5559cbedd8b42a93208b4890/html5/thumbnails/68.jpg)
Alternative
The point divides the line segment joining and internally in the ratio 2 : 3.
Find the coordinates of the point B.
4,1P 8,1A yxB ,
![Page 69: 11X1 T05 01 division of an interval (2011)](https://reader036.vdocument.in/reader036/viewer/2022062405/5559cbedd8b42a93208b4890/html5/thumbnails/69.jpg)
Alternative
The point divides the line segment joining and internally in the ratio 2 : 3.
Find the coordinates of the point B.
4,1P 8,1A yxB ,
A B
![Page 70: 11X1 T05 01 division of an interval (2011)](https://reader036.vdocument.in/reader036/viewer/2022062405/5559cbedd8b42a93208b4890/html5/thumbnails/70.jpg)
Alternative
The point divides the line segment joining and internally in the ratio 2 : 3.
Find the coordinates of the point B.
4,1P 8,1A yxB ,
A BP 3
2
If P divides AB internally in the ratio 2 : 3
![Page 71: 11X1 T05 01 division of an interval (2011)](https://reader036.vdocument.in/reader036/viewer/2022062405/5559cbedd8b42a93208b4890/html5/thumbnails/71.jpg)
Alternative
The point divides the line segment joining and internally in the ratio 2 : 3.
Find the coordinates of the point B.
4,1P 8,1A yxB ,
A BP 3
2
If P divides AB internally in the ratio 2 : 3
Then B divides AP externally in the ratio 5 : 3
![Page 72: 11X1 T05 01 division of an interval (2011)](https://reader036.vdocument.in/reader036/viewer/2022062405/5559cbedd8b42a93208b4890/html5/thumbnails/72.jpg)
Alternative
The point divides the line segment joining and internally in the ratio 2 : 3.
Find the coordinates of the point B.
4,1P 8,1A yxB ,
A BP 3
2
If P divides AB internally in the ratio 2 : 3
Then B divides AP externally in the ratio 5 : 3
8,1 4,1
![Page 73: 11X1 T05 01 division of an interval (2011)](https://reader036.vdocument.in/reader036/viewer/2022062405/5559cbedd8b42a93208b4890/html5/thumbnails/73.jpg)
Alternative
The point divides the line segment joining and internally in the ratio 2 : 3.
Find the coordinates of the point B.
4,1P 8,1A yxB ,
A BP 3
2
If P divides AB internally in the ratio 2 : 3
Then B divides AP externally in the ratio 5 : 3
8,1 4,1
3:5
![Page 74: 11X1 T05 01 division of an interval (2011)](https://reader036.vdocument.in/reader036/viewer/2022062405/5559cbedd8b42a93208b4890/html5/thumbnails/74.jpg)
Alternative
The point divides the line segment joining and internally in the ratio 2 : 3.
Find the coordinates of the point B.
4,1P 8,1A yxB ,
A BP 3
2
If P divides AB internally in the ratio 2 : 3
Then B divides AP externally in the ratio 5 : 3
8,1 4,1
3:5
![Page 75: 11X1 T05 01 division of an interval (2011)](https://reader036.vdocument.in/reader036/viewer/2022062405/5559cbedd8b42a93208b4890/html5/thumbnails/75.jpg)
Alternative
The point divides the line segment joining and internally in the ratio 2 : 3.
Find the coordinates of the point B.
4,1P 8,1A yxB ,
A BP 3
2
If P divides AB internally in the ratio 2 : 3
Then B divides AP externally in the ratio 5 : 3
8,1 4,1
3:5
, B
![Page 76: 11X1 T05 01 division of an interval (2011)](https://reader036.vdocument.in/reader036/viewer/2022062405/5559cbedd8b42a93208b4890/html5/thumbnails/76.jpg)
Alternative
The point divides the line segment joining and internally in the ratio 2 : 3.
Find the coordinates of the point B.
4,1P 8,1A yxB ,
A BP 3
2
If P divides AB internally in the ratio 2 : 3
Then B divides AP externally in the ratio 5 : 3
8,1 4,1
3:5
, B2 2
![Page 77: 11X1 T05 01 division of an interval (2011)](https://reader036.vdocument.in/reader036/viewer/2022062405/5559cbedd8b42a93208b4890/html5/thumbnails/77.jpg)
Alternative
The point divides the line segment joining and internally in the ratio 2 : 3.
Find the coordinates of the point B.
4,1P 8,1A yxB ,
A BP 3
2
If P divides AB internally in the ratio 2 : 3
Then B divides AP externally in the ratio 5 : 3
8,1 4,1
3:5
, B2 2
1513
![Page 78: 11X1 T05 01 division of an interval (2011)](https://reader036.vdocument.in/reader036/viewer/2022062405/5559cbedd8b42a93208b4890/html5/thumbnails/78.jpg)
Alternative
The point divides the line segment joining and internally in the ratio 2 : 3.
Find the coordinates of the point B.
4,1P 8,1A yxB ,
A BP 3
2
If P divides AB internally in the ratio 2 : 3
Then B divides AP externally in the ratio 5 : 3
8,1 4,1
3:5
, B2 2
1513 4583
![Page 79: 11X1 T05 01 division of an interval (2011)](https://reader036.vdocument.in/reader036/viewer/2022062405/5559cbedd8b42a93208b4890/html5/thumbnails/79.jpg)
Alternative
The point divides the line segment joining and internally in the ratio 2 : 3.
Find the coordinates of the point B.
4,1P 8,1A yxB ,
A BP 3
2
If P divides AB internally in the ratio 2 : 3
Then B divides AP externally in the ratio 5 : 3
8,1 4,1
3:5
, B2 2
1513 4583
2
4,28
![Page 80: 11X1 T05 01 division of an interval (2011)](https://reader036.vdocument.in/reader036/viewer/2022062405/5559cbedd8b42a93208b4890/html5/thumbnails/80.jpg)
Alternative
The point divides the line segment joining and internally in the ratio 2 : 3.
Find the coordinates of the point B.
4,1P 8,1A yxB ,
A BP 3
2
If P divides AB internally in the ratio 2 : 3
Then B divides AP externally in the ratio 5 : 3
8,1 4,1
3:5
, B2 2
1513 4583
2
4,28
2,4
![Page 81: 11X1 T05 01 division of an interval (2011)](https://reader036.vdocument.in/reader036/viewer/2022062405/5559cbedd8b42a93208b4890/html5/thumbnails/81.jpg)
Type 4: Finding the ratio
The point divides the interval externally in the ratio k : 1.
If A is the point and B is the point , find the value of k.
8,3P
1991 Extension 1 HSC Q1c)
4,6 4,0
![Page 82: 11X1 T05 01 division of an interval (2011)](https://reader036.vdocument.in/reader036/viewer/2022062405/5559cbedd8b42a93208b4890/html5/thumbnails/82.jpg)
Type 4: Finding the ratio
The point divides the interval externally in the ratio k : 1.
If A is the point and B is the point , find the value of k.
8,3P
1991 Extension 1 HSC Q1c)
4,6
• Draw the endpoints, ratio and cross the same as usual
4,0
![Page 83: 11X1 T05 01 division of an interval (2011)](https://reader036.vdocument.in/reader036/viewer/2022062405/5559cbedd8b42a93208b4890/html5/thumbnails/83.jpg)
Type 4: Finding the ratio
The point divides the interval externally in the ratio k : 1.
If A is the point and B is the point , find the value of k.
8,3P
1991 Extension 1 HSC Q1c)
4,6
• Draw the endpoints, ratio and cross the same as usual
4,6 4,0
4,0
![Page 84: 11X1 T05 01 division of an interval (2011)](https://reader036.vdocument.in/reader036/viewer/2022062405/5559cbedd8b42a93208b4890/html5/thumbnails/84.jpg)
Type 4: Finding the ratio
The point divides the interval externally in the ratio k : 1.
If A is the point and B is the point , find the value of k.
8,3P
1991 Extension 1 HSC Q1c)
4,6
• Draw the endpoints, ratio and cross the same as usual
4,6 4,0
1:k
4,0
![Page 85: 11X1 T05 01 division of an interval (2011)](https://reader036.vdocument.in/reader036/viewer/2022062405/5559cbedd8b42a93208b4890/html5/thumbnails/85.jpg)
Type 4: Finding the ratio
The point divides the interval externally in the ratio k : 1.
If A is the point and B is the point , find the value of k.
8,3P
1991 Extension 1 HSC Q1c)
4,6
• Draw the endpoints, ratio and cross the same as usual
4,6 4,0
1:k
4,0
![Page 86: 11X1 T05 01 division of an interval (2011)](https://reader036.vdocument.in/reader036/viewer/2022062405/5559cbedd8b42a93208b4890/html5/thumbnails/86.jpg)
Type 4: Finding the ratio
The point divides the interval externally in the ratio k : 1.
If A is the point and B is the point , find the value of k.
8,3P
1991 Extension 1 HSC Q1c)
4,6
• Draw the endpoints, ratio and cross the same as usual
4,6 4,0
1:k
4,0
• 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
![Page 87: 11X1 T05 01 division of an interval (2011)](https://reader036.vdocument.in/reader036/viewer/2022062405/5559cbedd8b42a93208b4890/html5/thumbnails/87.jpg)
Type 4: Finding the ratio
The point divides the interval externally in the ratio k : 1.
If A is the point and B is the point , find the value of k.
8,3P
1991 Extension 1 HSC Q1c)
4,6
• Draw the endpoints, ratio and cross the same as usual
4,6 4,0
1:k
10613
k
k
4,0
• 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
![Page 88: 11X1 T05 01 division of an interval (2011)](https://reader036.vdocument.in/reader036/viewer/2022062405/5559cbedd8b42a93208b4890/html5/thumbnails/88.jpg)
Type 4: Finding the ratio
The point divides the interval externally in the ratio k : 1.
If A is the point and B is the point , find the value of k.
8,3P
1991 Extension 1 HSC Q1c)
4,6
• Draw the endpoints, ratio and cross the same as usual
4,6 4,0
1:k
10613
k
k
4,0
• 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
633 k
![Page 89: 11X1 T05 01 division of an interval (2011)](https://reader036.vdocument.in/reader036/viewer/2022062405/5559cbedd8b42a93208b4890/html5/thumbnails/89.jpg)
Type 4: Finding the ratio
The point divides the interval externally in the ratio k : 1.
If A is the point and B is the point , find the value of k.
8,3P
1991 Extension 1 HSC Q1c)
4,6
• Draw the endpoints, ratio and cross the same as usual
4,6 4,0
1:k
10613
k
k
4,0
• 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
633 k
393
kk
![Page 90: 11X1 T05 01 division of an interval (2011)](https://reader036.vdocument.in/reader036/viewer/2022062405/5559cbedd8b42a93208b4890/html5/thumbnails/90.jpg)
Type 4: Finding the ratio
The point divides the interval externally in the ratio k : 1.
If A is the point and B is the point , find the value of k.
8,3P
1991 Extension 1 HSC Q1c)
4,6
• Draw the endpoints, ratio and cross the same as usual
4,6 4,0
1:k
10613
k
k
1:3ratioin theexternally divides ABP
4,0
• 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
633 k
393
kk
![Page 91: 11X1 T05 01 division of an interval (2011)](https://reader036.vdocument.in/reader036/viewer/2022062405/5559cbedd8b42a93208b4890/html5/thumbnails/91.jpg)
Type 4: Finding the ratio
The point divides the interval externally in the ratio k : 1.
If A is the point and B is the point , find the value of k.
8,3P
1991 Extension 1 HSC Q1c)
4,6
• Draw the endpoints, ratio and cross the same as usual
4,6 4,0
1:k
10613
k
k
1:3ratioin theexternally divides ABP
4,0
• 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
633 k
393
kk
Exercise 5A;1ad, 2ad,
3 i, iii in all, 4ace,5 i, ii ace, 6bd,
8, 9, 11, 13b, 16,17, 20, 21, 23, 24