Partitioning of Line Segments

1-VariableAnd

2-Variable

2 1 1( )a

x x xa b

Find a Point that Partitions a Segment in a Given Ratio a:b

(1 variable)

2 1 1( )

ax x x

a b

a: the first part of the ratiob: the second part of the ratioX1: first x value givenX2: second x value given

Partition – 1 Variable

A is at 1, and B is at 7.Find the point, T, so that T partitions A to B in a 2:1 ratio.

a:b:

Partition – 1 Variable

A is at -6 and B is at 4.Find the point, T, so that T is A to B in a 2:3 ratio.

a:b:

Find a Point that Partitions a Segment in a Given Ratio a:b

2 Variables

2 1 1 2 1 1,a a

x x x y y ya b a b

2 1 1 2 1 1,a a

x x x y y ya b a b

a: the first part of the ratiob: the second part of the ratioX1: first x-value given y1: first y-value X2: second x-value given y2: second y-value

Find the coordinates of P along the directed line segment AB so that the ratio of AP to PB is 3:2.

In order to divide the segment in the ratio of 3 to 2, think of dividing the segment into 3 + 2 or 5 congruent pieces.

A(3, 4), B(6, 10); 3 to 2.

To find the coordinates of point P…

1 2 1 2 1 1,

a ax x x y y y

a b a b

3 33 3, 6 4

5 5

4.8, 7.6

3 36 3 3, 10 4 4

5 5

a:b:

Example 1: Find the coordinates of point P along the directed line segment AB so that AP to PB is the given ratio. A(1, 3), B(8, 4); 4 : 1.

2 1 1 2 1 1, a a

x x x y y ya b a b

4 47 1, 1 3

5 5

6.6, 3.8

a:b:

4 48 1 1, 4 3 3

5 5

Example 2: Find the coordinates of point P along the directed line segment AB so that AP to PB is the given ratio. A(-2, 1), B(4, 5); 3 to 7.

2 1 1 2 1 1, a a

x x x y y ya b a b

3 36 2, 4 1

10 10

0.2, 2.2

a:b:

3 34 2 2, 5 1 1

10 10

Example 3: Find the coordinates of point P along the directed line segment AB so that AP to PB is the given ratio. A(8, 0), B(3, -2); 1 : 4.

2 1 1 2 1 1, a a

x x x y y ya b a b

1 15 8, 2 0

5 5

7, 0.4

a:b:

1 13 8 8, 2 0 0

5 5

Example 4: Find the coordinates of point P along the directed line segment AB so that AP to PB is the given ratio. A(-2, -4), B(6, 1); 3 to 2.

2 1 1 2 1 1, a a

x x x y y ya b a b

3 38 2, 5 4

5 5

2.8, 1

a:b:

3 36 2 2, 1 4 4

5 5

CW/HW