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Connecting Algebra and Geometry Through Coordinates

Position of Points on a Line Segment

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.

Try your knowledge on these topics.
a Consider the given right triangles, with two parallel legs.
Right Triangle
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
Scale Factor
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
Dilation
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

Directed Line Segment
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.

Directed Line Segment

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

Directed Line Segment

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

Directed Line Segment

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

Triangle

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

Triangle
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.
Evaluate
The coordinates of were found. By following the same procedure, the coordinates of can also be found.
Formula:
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.

Departure Time


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.
Evaluate
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:
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.

Choo Choo

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 A. Because each station is equidistant — the same distance to eachother — station A 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 A can be found.
Evaluate
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:
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.
Evaluate
Therefore, the coordinates of the point at which Dylan would have walked miles are
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