**Using Slope to Partition A Segment**

Ratio

A ratio is a comparison of two quantities.

Find the coordinates of the point P that lies along the directed line segment from

A

(3, 4) to

B

(6, 10) and partitions the segment in the ratio 3 to 2.

**The Challenge**

**Can you find the point on a line segment that partitions the segment into a given ratio?**

Step 2: Determine Delta of x and y

Step 1: Convert the Ratio

The ratio of

a

to

b

can be expressed as:

What is the ratio of...

Stars to hearts?

Hearts to stars?

Stars to total shapes?

Hearts to total shapes?

In a coordinate plane, slope is the ratio of

rise

to

run

.

Slope of a Straight Line

**a**

:

b

or

:

b

or

**a**

**b**

Step 3: Determine Increase in x and y

Step 4: Calculate the Location of Point P

To determine how much to increase

x

and

y

, the delta of

x

and the delta of

y

must each be multiplied by the converted ratio.

The last step is to add the amount of change in

x

and

y

determined in Step 3 to the original

x

and

y

value of the first endpoint.

Step 1:

Convert the ratio

Step 2:

Determine delta of

x

and

y

Step 3:

Determine the increase in

x

and

y

Step 4:

Calculate the new point

Steps for Partitioning a Segment

The Problem

The ratio of 3 to 2 must be first converted into a percent

numerator

numerator + denominator

**3**

**3 + 2**

**3**

**5**

**=**

**=**

Remember, delta is the change or difference in values. You will need to find the delta of

x

and the delta of

y

separately.

x

= 6 - 3 = 3

A directed line segment is a line segment that has a direction associated with it. It moves from one endpoint to the other.

Original Coordinates:

A

(3, 4) to

B

(6, 10)

Original Coordinates:

A

(3, 4) to

B

(6, 10)

y

= 10 - 4 = 6

The triangle is the symbol used to represent delta.

y

Increase in

y

:

3

5

converted ratio

6

=

18

5

= 3.6

x

Increase in

x

:

3

5

converted ratio

3

=

9

5

= 1.8

x

value for Point P

3 + 1.8 = 4.8

Starting

x

value

Increase in

x

+

y

value for Point P

4 + 3.6 = 7.6

Starting

y

value

Increase in

y

+

Coordinates of Point P

The directed line segment from

A

(3, 4) to

B

(6, 10) is partitioned at the ratio of 3 to 2 at point

P

(4.8, 7.6)

.