3. Position of Points on a Line Segment
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Chapter 4
3.
Position of Points on a Line Segment
| Student Learning Objectives: |
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Why
This lesson delves into the concept of a directed line segment and how a point on this segment can partition it in a specific ratio. Through various examples, such as determining the position of a point that divides a segment in a given ratio, the lesson provides insights into the practical applications of these mathematical concepts. One example involves a railroad company planning to construct a railroad with equidistant train stations, showcasing the real-world relevance of partitioning line segments. Another scenario discusses the journey of two individuals traveling between cities and how to determine their positions based on the distance they've traveled. Overall, the lesson offers a blend of theoretical knowledge and practical applications, making it easier for learners to grasp and apply these concepts in various situations.
Lesson Settings & Tools
| | 8 Theory slides |
| | 10 Exercises - Grade E - A |
| | Each lesson is meant to take 1-2 classroom sessions |
Position of Points on a Line Segment
Slide of 8
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.
Given that vertices A, B, and C are collinear and BE is parallel to CD, choose the correct similarity theorem that proves △ AEB ~ △ BDC.
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b In the following diagram, △ ABE is similar to △ ACD.
Given that AB=5, AE=4, and CD=6, identify the scale factor of △ ABE to △ ACD and find the length of AC.
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c Given two right triangles, △ ABC is the dilation of △ ADE by a scale factor of 52.
If the hypotenuse of △ ADE is 4, find the hypotenuse of △ ABC.
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Challenge
Investigating a Partitioned Line Segment
Dylan's home is 4.5 miles away from Big Apple Circus. Dylan walks from his home to the circus to watch the flying trapeze performance.
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Example
Partitioning a Directed Line Segment
In a math test, Emily is asked to find the position of point M that partitions the directed line segment KL in the ratio 3:2.
Help her identify the coordinates of M.
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Hint
Draw a right triangle with hypotenuse KM and another right triangle with hypotenuse KL.
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, KL is divided into 3 and 2 parts, represented by KM and ML respectively, out of 5 parts in total.
Notice that ∠ KNM and ∠ KPL are both right angles, and therefore they are congruent angles. Additionally, by the Reflexive Property of Congruence, it can be said that ∠ K is congruent to itself. ∠ KNM ≅ ∠ KPL and ∠ K ≅ ∠ K Therefore, by the Angle-Angle Similarity Theorem, it can be concluded that △ KMN and △ KLP are similar triangles. Furthermore, since KM represents 35 of the length of the directed line segment KL, it can be said that △ KMN is a dilation of △ KLP by a scale factor k of 35. k= 3/5 By the composition of the triangles, points M and N are on the same vertical line segment. Therefore, by finding the x-coordinate of N, the x-coordinate of point M can be identified. As derived from the diagram, the length of KP is 8 units. Therefore, KN can be found by multiplying 8 by the scale factor k. KN= 3/5* 8 ⇔ KN= 4.8 Thereby, point N is located 4.8 units away from point K in the positive horizontal direction. As a result, the x- coordinate of point N is - 4+ 4.8= 0.8. This means that N is located at ( 0.8, 1).
Consequently, the x-coordinate of point M is also 0.8. To find its y-coordinate, MN should be found. In the diagram, it can be seen that the length of LP is 4 units. With this information, MN can be found by multiplying 4 by the scale factor. MN= 3/5 * 4 ⇔ MN= 2.4 Point M is located 2.4 units away from point N in the positive vertical direction. Since the y-coordinate of N is 1 and 1+ 2.4 is equal to 3.4, point M is located at ( 0.8, 3.4).
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Discussion
General Method for Partitioning a Directed Line Segment
The approach used to find the coordinates of point M can be generalized algebraically. Suppose that M divides the directed line segment from A(x_1,y_1) to B(x_2,y_2) in the ratio a:b.
M ( x_1+ k( x_2- x_1), y_1+ k( y_2- y_1) )
⇕
M ( x_1 + a/a + b( x_2 - x_1), y_1 + a/a + b( y_2 - y_1) )
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Example
Finding Points on a Triangle's Perimeter
In △ PQR, point S partitions PQ from point P to point Q in the ratio 3:7. Similarly, point T partitions RP from point R to point P in the same ratio. With this information, Heichi is trying to find the positions of points S and T.
Given the endpoints of the directed line segments, help Heichi determine the coordinates of S and T.
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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 PQ and RP.
Next, the scale factor k can be found by using the formula for the scale factor of a dilation. k&= 3/3+7 ⇔ k= 0.3 Now, the coordinates of S can be found by substituting the coordinates of the endpoints of PQ into the formula for the point that partitions the directed line segment. Note that S divides PQ from point P to point Q. Therefore, P( -2, 5) and Q( -4, 1) will be substituted for ( x_1, y_1) and ( x_2, y_2), respectively.
S(x_1+k(x_2-x_1),y_1+k(y_2-y_1) )
SubstituteValues
Substitute values
S ( -2+ 0.3( -4-( -2)), 5+ 0.3( 1- 5) )
▼
Evaluate
SubNeg
a-(- b)=a+b
S (-2+0.3(-2),5+0.3(1-5) )
SubTerm
Subtract term
S (-2+0.3(-2),5+0.3(-4) )
MultPosNeg
a(- b)=- a * b
S (-2+(- 0.6),5+(- 1.2))
AddNeg
a+(- b)=a-b
S (-2.6,3.8)
The coordinates of S were found. By following the same procedure, the coordinates of T can also be found.
| Formula: ( x_1+k(x_2-x_1), y_1+k(y_2-y_1)) | ||||
|---|---|---|---|---|
| Segment | Endpoints | Scale Factor | Substitute | Simplify |
| PQ | P( -2, 5) and Q( -4, 1) | 0.3 | S ( -2+ 0.3( -4-( -2)), 5+ 0.3( 1- 5) ) | S (-2.6,3.8) |
| RP | R( 5, 1) and P( -2, 5) | 0.3 | T ( 5+ 0.3( -2- 5), 1+ 0.3( 5- 1) ) | T (2.9,2.2) |
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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.
| City | Longitude (x) | Latitude (y) |
|---|---|---|
| Madrid | ≈ - 3.70 | ≈ 40.42 |
| Warsaw | ≈ 21.01 | ≈ 52.23 |
The distance between the cities is approximately 2000 kilometers. Give the coordinates of the points at which they would have flown 500 kilometers. Round the coordinates to the nearest hundredth.
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Hint
Use the formula for identifying the position of a point on a directed line segment.
Solution
The coordinates of points V and M 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 2000 kilometers. It is given that each passenger has flown 500 kilometers. With this information, the scale factor of the dilation can be identified.
k= 500/2000 ⇔ k= 0.25
The scale factor for both routes is 0.25. From here, the coordinates of V 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, ( -3.70, 40.42) and ( 21.01, 52.23) will be substituted for ( x_1, y_1) and ( x_2, y_2), respectively.
V(x_1+k(x_2-x_1),y_1+k(y_2-y_1) )
SubstituteValues
Substitute values
V ( -3.70+ 0.25( 21.01-( -3.70)), 40.42+ 0.25( 52.23- 40.42) )
V (2.48,43.37)
By following the same procedure, the coordinates of M 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: ( x_1+k(x_2-x_1), y_1+k(y_2-y_1)) | ||||
|---|---|---|---|---|
| Route | Endpoints | Scale Factor | Substitute | Simplify |
| Madrid to Warsaw | ( -3.70, 40.42) and ( 21.01, 52.23) | 0.25 | V ( -3.70+ 0.25( 21.01-( -3.70)), 40.42+ 0.25( 52.23- 40.42) ) | V (2.48,43.37) |
| Warsaw to Madrid | ( 21.01, 52.23) and ( -3.70, 40.42) | 0.25 | M ( 21.01+ 0.25( -3.70- 21.01), 52.23+ 0.25( 40.42- 52.23) ) | M (14.83,49.28) |
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.
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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.
| City | Longitude (x) | Latitude (y) |
|---|---|---|
| Houston | ≈ - 95.37 | ≈ 29.76 |
| Washington D.C. | ≈ -77.04 | ≈ 38.91 |
Find the coordinates of each station. Round them to the nearest hundredth.
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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 1:4. By using this information, the scale factor of the dilation can be found.
k= 1/1+4 ⇔ k= 0.2
Now, by substituting the scale factor and the coordinates of the cities into the formula, the coordinates of station A can be found.
A(x_1+k(x_2-x_1),y_1+k(y_2-y_1) )
SubstituteValues
Substitute values
A ( -95.37+ 0.2( -77.04-( -95.37)), 29.76+ 0.2( 38.91- 29.76) )
A (-91.70,31.59)
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 |
|---|---|---|---|
| A | 1:4 | k=1/1+4 | k= 0.2 |
| B | 2:3 | k=2/2+3 | k= 0.4 |
| C | 3:2 | k=3/3+2 | k= 0.6 |
| D | 4:1 | k=4/4+1 | k= 0.8 |
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 ( -95.37, 29.76) and ( -77.04, 38.91).
| Formula: ( x_1+k(x_2-x_1), y_1+k(y_2-y_1)) | |||
|---|---|---|---|
| Station | Scale Factor | Substitute | Simplify |
| A | 0.2 | A ( -95.37+ 0.2( -77.04-( -95.37)), 29.76+ 0.2( 38.91- 29.76) ) | A(-91.70,31.59) |
| B | 0.4 | B ( -95.37+ 0.4( -77.04-( -95.37)), 29.76+ 0.4( 38.91- 29.76) ) | B(-88.04,33.42) |
| C | 0.6 | C ( -95.37+ 0.6( -77.04-( -95.37)), 29.76+ 0.6( 38.91- 29.76) ) | C(-84.37,35.25) |
| D | 0.8 | D ( -95.37+ 0.8( -77.04-( -95.37)), 29.76+ 0.8( 38.91- 29.76) ) | D(-80.71,37.08) |
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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 3 miles of the 4.5-mile distance were asked to be found.
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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 3 miles partitions the road in the ratio 2:1.
3/1.5 ⇔ 2:1
With this information, the scale factor of the dilation can be determined.
k= 2/2+1 ⇔ k= 2/3
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, ( 2, 5) and ( 6.5, 2) will be substituted for ( x_1, y_1) and ( x_2, y_2), respectively.
D(x_1+k(x_2-x_1),y_1+k(y_2-y_1) )
SubstituteValues
Substitute values
D ( 2+ 2/3( 6.5- 2), 5+ 2/3( 2- 5) )
▼
Evaluate
SubTerms
Subtract terms
D (2+2/3(4.5),5+2/3(-3) )
MultPosNeg
a(- b)=- a * b
D (2+2/3(4.5),5-2/3(3) )
MoveRightFacToNum
a/c* b = a* b/c
D (2+9/3,5-6/3 )
CalcQuot
Calculate quotient
D (2+3,5-2 )
AddSubTerms
Add and subtract terms
D (5,3)
Therefore, the coordinates of the point at which Dylan would have walked 3 miles are (5,3).
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Position of Points on a Line Segment
Exercises
Find the coordinates of point P along the directed line segment AB such that AP to PB is in the ratio of 4:1. Answer in exact form.
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A ratio of 4 to 1 means that if we partition AB into five congruent segments, four of these segments will belong to AP, and one segment belongs to PB. Let's illustrate this case in our coordinate plane.
To move from A to P, we need to travel four-fifths of the horizontal and vertical distance from A to B. Let's first measure these distances.
Notice that P is above and to the right of A. Therefore, to obtain the coordinates of P, we have to add 45 of the horizontal and vertical distance, between A and B, to the x- and y-coordinates of A( 1, 3). x_P:& 1 +4/5( 7)=33/5 [1em] y_P:& 3 +4/5( 1)=19/5 Therefore, the coordinates of P are ( 335, 195).
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Solution
Find the coordinates of point P along the directed line segment AB such that AP to PB is in the ratio 2:3. Answer in exact form.
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A ratio of 2 to 3 means that if we partition AB into five congruent segments, two of these will belong to AP, and three will belong to PB. Let's illustrate this in our coordinate plane.
To move from A to P, we need to travel two-fifths of the vertical and horizontal distance from A to B. Let's measure these distances.
Examining the diagram, we can determine that P is below and to the left of A. This means that to obtain the coordinates of P, we could subtract 25 of the horizontal and vertical distances, between A and B, from the x- and y-coordinates of A( 6, 1), respectively. x_P:& 6 -2/5( 8)=14/5 [1em] y_P:& 1 -2/5( 5)=-1 Therefore, the coordinates of P are ( 145,-1).
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Solution
What is the ratio of NL to NM?
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To determine the ratio of NL to NM, we need to know both of these distances. This requires us to know their coordinates. From the diagram, we can identify the coordinates of both N and M.
Now we can calculate the length of NM using the Distance Formula.
To determine the length of NL, we need to know the coordinates of L. From the diagram, we see that it has an x-coordinate of 1. To find its y-coordinate, we will add two right triangles to the diagram.
Since these triangles are both right triangles and share an angle, they are similar by the Angle-Angle Similarity Theorem. With this information, we can write an equation. y_L/5=4/7 Let's solve for y_L.
Now we can calculate the length of NL using the Distance Formula.
Now we can determine the ratio of NL to NM.
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Ignacio and his family took the train to see the circus. There is a 40 minute walk from the train station to reach the circus, as illustrated in the diagram. After how many minutes would you expect Ignacio and his family to reach the circus if they walk at a constant rate.
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To determine the number of minutes it takes to walk from the train station to the factory, we can use the ratio of the distances TF to TC. Let's find the ratio by drawing two right triangles in the diagram that share a vertex in T. Since T, F and C all fall on lattice points, we can determine the lengths of the legs.
Using the Pythagorean Theorem, we can calculate the distances TF and TC.
Let's also calculate TC.
If we divide TF by TC and multiply this by 40 minutes, we can determine the number of minutes it will take to walk from the train station to the factory.
It takes 16 minutes for Igancio and his family to reach the factory by walking from the train station.
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Position of Points on a Line Segment
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