11 X1 T05 01 Division Of An Interval

Coordinate Geometry

Coordinate GeometryDistance Formula


2 2

2 1 2 1d x x y y


2 2

2 1 2 1d x x y y

e.g. Find the distance between (–1,3) and (3,5)


2 2

2 1 2 1d x x y y


2 25 3 3 1d


2 2

2 1 2 1d x x y y


2 25 3 3 1d

2 22 4

20

2 5 units


2 2

2 1 2 1d x x y y


2 25 3 3 1d

2 22 4

20

2 5 units

The distance formula is finding the length of the hypotenuse,

using Pythagoras


2 2

2 1 2 1d x x y y


2 25 3 3 1d

2 22 4

20

2 5 units

Midpoint Formula


using Pythagoras


2 2

2 1 2 1d x x y y


2 25 3 3 1d

2 22 4

20

2 5 units

Midpoint Formula

1 2 1 2,2 2

x x y yM


using Pythagoras


2 2

2 1 2 1d x x y y


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)


using Pythagoras


2 2

2 1 2 1d x x y y


2 25 3 3 1d

2 22 4

20

2 5 units

Midpoint Formula

1 2 1 2,2 2

x x y yM


3 2 4 1,

2 2M


using Pythagoras


2 2

2 1 2 1d x x y y


2 25 3 3 1d

2 22 4

20

2 5 units

Midpoint Formula

1 2 1 2,2 2

x x y yM


3 2 4 1,

2 2M

1 5,

2 2


using Pythagoras


2 2

2 1 2 1d x x y y


2 25 3 3 1d

2 22 4

20

2 5 units

Midpoint Formula

1 2 1 2,2 2

x x y yM


3 2 4 1,

2 2M

1 5,

2 2


using Pythagoras

The midpoint formula averages the x and y values

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




X YA




X YA

A divides XY internally in the ratio m:n




X YA


m

n




X YA


m

n

OR




X YA


m

n

ORA divides YX internally in the ratio n:m




X YA

X Y


m

n





X YA

X Y A


m

n





X YA

X Y A


m

n


A divides XY externally in the ratio m:n




X YA

X Y A


m

n


A divides XY externally in the ratio m:n

m

n

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



4,3


6,5

• Write down the endpoints of the interval in the same order as they are mentioned.



4,3


6,5


4,3 6,5



4,3


6,5


4,3 6,5

• Write down the ratio.



4,3


6,5


4,3 6,5


3:1



4,3


6,5


4,3 6,5


3:1

• Draw a cross joining the ratio to the two points



4,3


6,5


4,3 6,5


3:1


• Set up your answer by drawing a set of parentheses with two vinculums separated by a comma



4,3


6,5


4,3 6,5


3:1



,

P



4,3


6,5


4,3 6,5


3:1



,

P

• Add the numbers in the ratio together to get the denominator



4,3


6,5


4,3 6,5


3:1



,

P


4 4



4,3


6,5


4,3 6,5


3:1



,

P


4 4

• Multiply along the cross and add to get the numerator



4,3


6,5


4,3 6,5


3:1



,

P


4 4


5133



4,3


6,5


4,3 6,5


3:1



,

P


4 4


5133 6143



4,3


6,5


4,3 6,5


3:1



,

P


4 4


5133 6143

418

,44



4,3


6,5


4,3 6,5


3:1



,

P


4 4


5133 6143

418

,44

29

,1

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


2,9



1,3


2,9

• Done exactly the same as internal division, except make one of the numbers in the ratio negative



1,3


2,9


1,3 2,9



1,3


2,9


1,3 2,9

2:5



1,3


2,9


1,3 2,9

2:5

,

P



1,3


2,9


1,3 2,9

2:5

,

P3 3



1,3


2,9


1,3 2,9

2:5

,

P3 3

9532



1,3


2,9


1,3 2,9

2:5

,

P3 3

9532 2512



1,3


2,9


1,3 2,9

2:5

,

P3 3

9532 2512

312

,3

39



1,3


2,9


1,3 2,9

2:5

,

P3 3

9532 2512

312

,3

39

4,13

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 ,




4,1P


8,1A

• Draw the endpoints, ratio and cross the same as previously

yxB ,




4,1P


8,1A


8,1 yx,

yxB ,




4,1P


8,1A


8,1 yx,

3:2

yxB ,




4,1P


8,1A


8,1 yx,

3:2

yxB ,

• Create the fraction for the x value and equate it with the known value




4,1P


8,1A


8,1 yx,

3:25

2131

x

yxB ,





4,1P


8,1A


8,1 yx,

3:25

2131

x

yxB ,


x235




4,1P


8,1A


8,1 yx,

3:25

2131

x

yxB ,


x235

4

82

x

x




4,1P


8,1A


8,1 yx,

3:25

2131

x

yxB ,


• Repeat for the y value

x235

4

82

x

x




4,1P


8,1A


8,1 yx,

3:25

2131

x5

2834

y

yxB ,



x235

4

82

x

x




4,1P


8,1A


8,1 yx,

3:25

2131

x5

2834

y

yxB ,



x235

4

82

x

xy22420




4,1P


8,1A


8,1 yx,

3:25

2131

x5

2834

y

yxB ,



x235

4

82

x

xy22420

2

42

y

y




4,1P


8,1A


8,1 yx,

3:25

2131

x5

2834

y

2,4 B

yxB ,



x235

4

82

x

xy22420

2

42

y

y

Alternative



4,1P 8,1A yxB ,

Alternative



4,1P 8,1A yxB ,

A B

Alternative



4,1P 8,1A yxB ,

A BP 3

2

If P divides AB internally in the ratio 2 : 3

Alternative



4,1P 8,1A yxB ,

A BP 3

2


Then B divides AP externally in the ratio 5 : 3

Alternative



4,1P 8,1A yxB ,

A BP 3

2



8,1 4,1

Alternative



4,1P 8,1A yxB ,

A BP 3

2



8,1 4,1

3:5

Alternative



4,1P 8,1A yxB ,

A BP 3

2



8,1 4,1

3:5

,

B

Alternative



4,1P 8,1A yxB ,

A BP 3

2



8,1 4,1

3:5

,

B2 2

Alternative



4,1P 8,1A yxB ,

A BP 3

2



8,1 4,1

3:5

,

B2 2

1513

Alternative



4,1P 8,1A yxB ,

A BP 3

2



8,1 4,1

3:5

,

B2 2

1513 4583

Alternative



4,1P 8,1A yxB ,

A BP 3

2



8,1 4,1

3:5

,

B2 2

1513 4583

24

,28

Alternative



4,1P 8,1A yxB ,

A BP 3

2



8,1 4,1

3:5

,

B2 2

1513 4583

24

,28

2,4

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


4,6 4,0




8,3P


4,6

• Draw the endpoints, ratio and cross the same as usual

4,0




8,3P


4,6


4,6 4,0

4,0




8,3P


4,6


4,6 4,0

1:k

4,0




8,3P


4,6


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




8,3P


4,6


4,6 4,0

1:k

1061

3

k

k

4,0





8,3P


4,6


4,6 4,0

1:k

1061

3

k

k

4,0


633 k




8,3P


4,6


4,6 4,0

1:k

1061

3

k

k

4,0


633 k

3

93

k

k




8,3P


4,6


4,6 4,0

1:k

1061

3

k

k

1:3 ratio in the externally divides ABP

4,0


633 k

3

93

k

k




8,3P


4,6


4,6 4,0

1:k

1061

3

k

k

1:3 ratio in the externally divides ABP

4,0


633 k

3

93

k

k

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

11 X1 T05 01 Division Of An Interval

Education

Transcript of 11 X1 T05 01 Division Of An Interval