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A point on a directed line segment partitions the segment in a specific ratio. In this lesson, given the segment's endpoints and ratio, the point's coordinates will be identified.

Catch-Up and Review

Here are a few recommended readings before getting started with this lesson.

Congruence
Similarity
Dilation
Right Triangles

Try your knowledge on these topics.

a Consider the given right triangles, with two parallel legs.

Given that vertices

and

are collinear and

is parallel to

choose the correct similarity theorem that proves

b In the following diagram,

is similar to

Given that

and

identify the scale factor of

to

and find the length of

c Given two right triangles,

is the dilation of

by a scale factor of

If the hypotenuse of

is

find the hypotenuse of

Challenge

Investigating a Partitioned Line Segment

Dylan's home is

miles away from Big Apple Circus. Dylan walks from his home to the circus to watch the flying trapeze performance.

Point on a Directed Line Segment

What are the coordinates of the point at which Dylan would have walked

miles?

Example

Partitioning a Directed Line Segment

In a math test, Emily is asked to find the position of point that partitions the directed line segment in the ratio

Help her identify the coordinates of

Hint

Draw a right triangle with hypotenuse and another right triangle with hypotenuse

Solution

The position of a point that partitions a directed line segment in a given ratio can be found by using a dilation. In this case, two right triangles can be drawn to identify the dilation. According to the given ratio, is divided into and parts, represented by and respectively, out of parts in total.

Notice that

and

are both right angles, and therefore they are congruent angles. Additionally, by the Reflexive Property of Congruence, it can be said that

is congruent to itself.

Therefore, by the Angle-Angle Similarity Theorem, it can be concluded that

and

are similar triangles. Furthermore, since

represents

of the length of the directed line segment

it can be said that

is a dilation of

by a scale factor

of

By the composition of the triangles, points

and

are on the same vertical line segment. Therefore, by finding the

coordinate of

the

coordinate of point

can be identified. As derived from the diagram, the length of

is

units. Therefore,

can be found by multiplying

by the scale factor

Thereby, point

is located

units away from point

in the positive horizontal direction. As a result, the

coordinate of point

is

This means that

is located at

Consequently, the

coordinate of point

is also

To find its

coordinate,

should be found. In the diagram, it can be seen that the length of

is

units. With this information,

can be found by multiplying

by the scale factor.

Point

is located

units away from point

in the positive vertical direction. Since the

coordinate of

is

and

is equal to

point

is located at

Discussion

General Method for Partitioning a Directed Line Segment

The approach used to find the coordinates of point

can be generalized algebraically. Suppose that

divides the directed line segment from

to

in the ratio

Point on a Directed Line Segment

Since

divides

from

to

point

is said to be the center of dilation. Next, two right triangles will be considered to show the dilation. The directed distances from

to

in the

and

directions can be identified by subtracting the endpoints of the segment.

Point on a Directed Line Segment

Since

and

are right angles, they are congruent to each other. Furthermore, by the Reflexive Property of Congruence, it can be stated

is congruent to itself. With this information, by the Angle-Angle Similarity Theorem,

and

are similar triangles.

The next step to identify the position of

is to express the ratio

as the scale factor of the dilation. Since the hypotenuses of

and

are

and

respectively, the scale factor

is the quotient between

and

Because

and

are similar triangles,

and

can be found by multiplying

and

respectively, by the scale factor

Finally, the coordinates of

can be found by adding the resulting distances to the coordinates of the center of dilation.

Example

Finding Points on a Triangle's Perimeter

In point partitions from point to point in the ratio Similarly, point partitions from point to point in the same ratio. With this information, Heichi is trying to find the positions of points and

Given the endpoints of the directed line segments, help Heichi determine the coordinates of and

Hint

Use the formula for identifying the position of a point on a directed line segment.

Solution

First, the given ratio will be expressed on and

Next, the scale factor

can be found by using the formula for the scale factor of a dilation.

Now, the coordinates of

can be found by substituting the coordinates of the endpoints of

into the formula for the point that partitions the directed line segment. Note that

divides

from point

to point

Therefore,

and

will be substituted for

and

respectively.

SubstituteValues

Substitute values

▼

Evaluate

Subtract term

The coordinates of

were found. By following the same procedure, the coordinates of

can also be found.

Formula: $\bm{\big( x_1+k(x_2-x_1), y_1+k(y_2-y_1)\big)}$
Segment	Endpoints	Scale Factor	Substitute	Simplify
	and
	and

Example

Partitioning Distances Between Cities on a Map

Vincenzo and Magdalena are two high school students. They are visiting different cities during their summer holidays. Vincenzo is currently traveling from Madrid to Warsaw, while Magdalena is traveling from Warsaw to Madrid.

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The approximated coordinates of the cities are given in the following table.

City	Longitude ()	Latitude ()
Madrid
Warsaw

The distance between the cities is approximately

kilometers. Give the coordinates of the points at which they would have flown

kilometers. Round the coordinates to the nearest hundredth.

Hint

Use the formula for identifying the position of a point on a directed line segment.

Solution

The coordinates of points

and

can be found by using the formula for identifying the position of a point on a directed line segment. The distance between Madrid and Warsaw is

kilometers. It is given that each passenger has flown

kilometers. With this information, the scale factor of the dilation can be identified.

The scale factor for both routes is

From here, the coordinates of

can be found by substituting the coordinates of the cities and the scale factor into the formula. Since Vincenzo is traveling from Madrid to Warsaw,

and

will be substituted for

and

respectively.

SubstituteValues

Substitute values

▼

Evaluate

Subtract term

Multiply

Add terms

Round to decimal place(s)

By following the same procedure, the coordinates of

can also be found. Be aware that Magdalena is traveling in the opposite direction of Vincenzo, from Warsaw to Madrid, while Vincenzo is traveling from Madrid to Warsaw.

Formula: $\bm{\big( x_1+k(x_2-x_1), y_1+k(y_2-y_1)\big)}$
Route	Endpoints	Scale Factor	Substitute	Simplify
Madrid to Warsaw	and
Warsaw to Madrid	and

Even if Vincenzo and Magdalena travel between the same cities, they are in different positions. The reason for their differing positions is because they are traveling in opposite directions. This reasoning would suggest, a point's position on a directed line segment depends on which endpoint is used as the center of dilation.

Example

Partitioning a Distance Into Congruent Segments

A railroad company is planning to construct a railroad from Houston to Washington D.C. They will place four train stations between the initial and the final station. Each station will be equidistant to each other, as the diagram below shows.

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The coordinates of the cities are given in a table.

City	Longitude ()	Latitude ()
Houston
Washington D.C.

Find the coordinates of each station. Round them to the nearest hundredth.

Hint

Use the formula for identifying the position of a point on a directed line segment.

Solution

Consider station

Because each station is equidistant — the same distance to eachother — station

partitions the railroad from Houston to Washington in the ratio

By using this information, the scale factor of the dilation can be found.

Now, by substituting the scale factor and the coordinates of the cities into the formula, the coordinates of station

can be found.

SubstituteValues

Substitute values

▼

Evaluate

Subtract term

Multiply

Add terms

Round to decimal place(s)

By following the same procedure, the coordinates of the other stations can also be found. First, the scale factors should be identified for each station.

Station	Ratio	Scale Factor	Simplify

From here, the positions of the other stations can be identified. Keep in mind that all stations go from Houston to Washington. Therefore, the endpoints are always and

Formula: $\bm{\big( x_1+k(x_2-x_1), y_1+k(y_2-y_1)\big)}$
Station	Scale Factor	Substitute	Simplify

Closure

Finding a Point on a Directed Line Segment

By using the formula learned in this lesson, the challenge presented at the beginning can be solved. Recall that the coordinates of the point at which Dylan would have walked

miles of the

mile distance were asked to be found.

Point on a Directed Line Segment

Hint

Determine the scale factor of the dilation. Use the formula for identifying the position of a point on a directed line segment.

Solution

Notice that the point at which Dylan stands after he walked

miles partitions the road in the ratio

With this information, the scale factor of the dilation can be determined.

By substituting the scale factor and the coordinates of the endpoints into the corresponding formula, Dylan's position can be found. Since he walks from his house to the circus,

and

will be substituted for

and

respectively.

SubstituteValues

Substitute values

▼

Evaluate

Subtract terms

MoveRightFacToNum

Calculate quotient

Add and subtract terms

Therefore, the coordinates of the point at which Dylan would have walked

miles are

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