CC Standard
G-GPE.6 Find the point on a directed line
Segment between two given points that partitions the segment in a given ratio.
At the end of this lesson, you should be able to answer the following question:
How do you find the point on a directed line segment that partitions the segment in a given ratio?

Directed Line Segment: has a starting point and a end point, a direction.
To partition a directed line segment is to divide it into two segments with a given ratio.



A ratio is a comparison of two quantities
The ratio of a to b can be expressed as: a : b or a / b or a b

Connor has a wallet with:
1-$20 bill
2- $10 bills
1- $5 bill
8-$1 bills
1) What is the ratio of $1 bills to $10 bills?
8:2 or 8/2 or 4/1 or 4:1
2) What is the ratio of $10 bills to the total number of bills in the wallet?
2:12 or 2/12 or 1/6 or 1:6

A 32 foot long piece of rope has a knot tied to divide the rope into a ratio of 1:1.
A
32 ft
Where should the knot be tied?
1. Divide the rope into 8 sections
2. 32ft divided by 8 is 4ft
3. There should be 1 unit in the first partition for every 1 unit in the 2 nd
4. If the ratio is 1:1, then there is the same amout on each side of the partition. 4:4 is the same as 1:1
5. The knot should be tied at 16 feet
6. A 1:1 ratio is the mid point

 Used to find the center of a line segment if you are given the coordinates of the 2 ends midpoint x
 x y
 y
,
2 2

• Find the midpoint between A(4,8) and B(1,12)
 S
(4 , 8 )
 A
(1, 12)
(5, 20)
 D
2 2
(2.5, 10)

S


Find the midpoint between:
1) A (-4, 5) and C(3, -4)
 
A
D
2
A

C
2 2
B ( .5, .5)

S


Find the midpoint between:
2) A (-3, -4) and C(6, -5)

A
D
2
A
 
C
(3, -9)
2 2
B (1.5, -4.5)

Ex. In segment AC
A is (4, 5) and the mid point is B(10, 12).
Find the endpoint C
We can still use SAD
2,
But we have to work it backwards and opposite.

1. In segment AC
A is (4, 5) and the mid point is B(10, 12).
Find the endpoint C
We can still use SAD
2,
But we have to work it backwards and
S
A
D
2
C ( , )
A ( 3, 4)
B ( 4,5)
M
2
S 
A


8 10
,
2 2
C ( 11,14)
1. Multiply m.pt by 2
2. Subtract the end
3. That is your end
4. Check it

2. In segment
AC
A is (6, -5) and the mid point is
B(-3, -4).
Find the endpoint C
S
M
2
S
A D
2
Remember, if they give you a midpt, use SAD, but backwards and opposite, also make sure you start with the mid point
B

( 3, 4)
A

C ( 12, 3)

1. What does this mean
?
 There are 3 units between A and P for every
1 unit between P and B

A 32 foot long piece of rope has a knot tied to divide the rope into a ratio of 3:5.
A
32 ft
Where should the knot be tied?
1. Divide the rope into 8 sections
2. 32ft divided by 8 is 4ft
3. There should be 3 units in the first partition for every 5 in the 2 nd
4. The knot should be tied at 12 feet

Finding a point that partitions a segment into a specific ratio
Given the points A(-1,2) and
B(7,8), find the coordinates of the point P on the directed line segment AB that partitions AB into the ratio 1:1
*Start with the end, B
Ratio of 1:1 is same as midpoint.
But watch what we
Do with the ratio…
2
1
1
B ( 7,8)
A ( -1,2)
B ( 7,8)
A ( -1,2)
( 6,10)
2 2
P ( 3 ,5)
S
A
D

Finding a point that partitions a segment into a specific ratio
Given the points A(-1,2) and
B(7,8), find the coordinates of the point P on the directed line segment AB that partitions AB into the ratio 1:3
*Start with the end, B
We can use SAD, but have to adjust it for the ratio.
4
1
3
B ( 7,8)
A ( -1,2)
B ( 7,8)
A ( -3,6)
( 4,14)
4 4
P ( 1,3.5)
S
A
D

Finding a point that partitions a segment into a specific ratio
We can use SAD, but have to adjust it for the ratio.
2. Find the point Q along the directed line segment from point
R(-3, 3) to point S(6,-3) that divides the segment into the ratio 2:3
S
2
3
S ( 6, 3)
R ( -3,3)
S ( 12, 6)
R ( -9, 9)
S
5
( 3, 3)
5 5
P ( .6, .6)
A
D