Explore Partitioning a Segment in a One-Dimensional Coordinate System
It takes just one number to specify an exact location on a number line. For this reason, a number line is sometimes called a one-dimensional coordinate system. The mile markers on a straight stretch of a highway turn that part of the highway into a one-dimensional coordinate system.
On a straight highway, the exit for Arthur Avenue is at mile marker 14. The exit for Collingwood Road is at mile marker 44. The state highway administration plans to put an exit for Briar Street at a point that is 2 _ 3 of the distance from Arthur Avenue to Collingwood Road. Follow these steps to determine where the new exit should be placed.
A Mark Arthur Avenue (point A) and Collingwood Road (point C) on the number line.
B What is the distance from Arthur Avenue to Collingwood Road? Explain.
C How far will the Briar Street exit be from Arthur Avenue? Explain.
D What is the mile marker number for the Briar Street exit? Why?
E Plot and label the Briar Street exit (point B) on the number line.
Module 12 641 Lesson 2
12.2 Subdividing a Segment in a Given Ratio
Essential Question: How do you find the point on a directed line segment that partitions the given segment in a given ratio?
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The highway administration also plans to put an exit for Dakota Lane at a point that divides the highway from Arthur Avenue to Collingwood Road in a ratio of 2 to 3. What is the mile marker number for Dakota Lane? Why? (Hint: Let the distance from Arthur Avenue to Dakota Lane be 2x and let the distance from Dakota Lane to Collingwood Road be 3x.)
Plot and label the Dakota Lane exit (point D) on the number line.
Reflect
1. How can you tell that the location at which you plotted point B is reasonable?
2. Would your answer in Step F be different if the exit for Dakota Lane divided the highway from Arthur Avenue to Collingwood Road in a ratio of 3 to 2? Explain.
Explain 1 Partitioning a Segment in a Two-Dimensional Coordinate System
A directed line segment is a segment between two points A and B with a specified direction, from A to B or from B to A. To partition a directed line segment is to divide it into two segments with a given ratio.
Example 1 Find the coordinates of the point P that divides the directed line segment from A to B in the given ratio.
A A (-8, -7) , B (8, 5) ; 3 to 1
Step 1 Write a ratio that expresses the distance of point P along the segment from A to B.
Point P is 3 _ 3 + 1 = 3 _ 4 of the distance from A to B.
Step 2 Find the run and the rise of the directed line segment.
run = 8 - (-8) = 16
rise = 5 - (-7) = 12
Module 12 642 Lesson 2
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Find the coordinates of the point P that divides the directed line segment from A to B in the given ratio.
5. A (-6, 5) , B (2, -3) ; 5 to 3 6. A (4, 2) , B (-6, -13) ; 3 to 2
Explain 2 Constructing a Partition of a Segment
Example 2 Given the directed line segment from A to B, construct the point P that divides the segment in the given ratio from A to B.
A 2 to 1
Step 1 Use a straightedge to draw ‾→ AC . The exact measure of the angle
is not important, but the construction is easiest for angles from about 30° to 60°.
Step 2 Place the compass point on A and draw an arc through ‾→ AC . Label
the intersection D. Using the same compass setting, draw an arc centered on D and label the intersection E. Using the same compass setting, draw an arc centered on E and label the intersection F.
Step 3 Use the straightedge to connect points B and F. Construct an angle congruent to ∠AFB with D as its vertex. Construct an angle congruent to ∠AFB with E as its vertex.
Step 4 The construction partitions _ AB into 3 equal parts. Label point P at
the point that divides the segment in the ratio 2 to 1 from A to B.
Module 12 644 Lesson 2
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Step 2 Place the compass point on A and draw an arc through ‾→ AC . Label the intersection D. Using the
same compass setting, draw an arc centered on D and label the intersection E. Using the same compass setting, draw an arc centered on E and label the intersection F. Using the same compass setting, draw an arc centered on F and label the intersection G.
Step 3 Use the straightedge to connect points B and G. Construct angles congruent to ∠AGB with D, E, and F as the vertices.
Step 4 The construction partitions _ AB into equal parts. Label point P at the point that divides the
segment in the ratio to from A to B.
Reflect
7. In Part A, why is _ EP is parallel to
_ FB ?
8. How can you use the Triangle Proportionality Theorem to explain why this construction method works?
Your Turn
Given the directed line segment from A to B, construct the point P that divides the segment in the given ratio from A to B.
9. 1 to 2 10. 3 to 2
Module 12 645 Lesson 2
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Elaborate
11. How is a one-dimensional coordinate system similar to a two-dimensional coordinate system? How is it different?
12. Is finding a point that is 4 _ 5 of the distance from point A to point B the same as finding a point that divides
_ AB in the ratio 4 to 5? Explain.
13. Essential Question Check-In What are some different ways to divide a segment in the ratio 2 to 1?
Evaluate: Homework and Practice
• Online Homework• Hints and Help• Extra Practice
A choreographer uses a number line to position dancers for a ballet. Dancers A and B have coordinates 5 and 23, respectively. In Exercises 1–4, find the coordinate for each of the following dancers based on the given locations.
1. Dancer C stands at a point that is 5 _ 6 of the 2. Dancer D stands at a point that is 1 _ 3 of the distance from Dancer A to Dancer B. distance from Dancer A to Dancer B.
3. Dancer E stands at a point that divides the line segment from Dancer A to Dancer B in a ratio of 2 to 1.
4. Dancer F stands at a point that divides the line segment from Dancer A to Dancer B in a ratio of 1 to 5.
Find the coordinates of the point P that divides the directed line segment from A to B in the given ratio.
5. A ( −3, −2 ) , B (12, 3) ; 3 to 2 6. A (−1, 5) , B (7, −3) ; 7 to 1
7. A (−1, 4) , (B-9, 0) ; 1 to 3 8. A (7, −3) , B (-7, 4) ; 3 to 4
Given the directed line segment from A to B, construct the point P that divides the segment in the given ratio from A to B.9. 3 to 1 10. 2 to 3
Module 12 647 Lesson 2
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A
B
A
B
20-20
J K L N
-10 0 10
M
11. 1 to 4 12. 4 to 1
Find the coordinate of the point P that divides each directed line segment in the given ratio.
13. from J to M; 1 to 9 14. from K to L; 1 to 1
15. from N to K; 3 to 5 16. from K to J; 7 to 11
17. Communicate Mathematical Ideas Leon constructed a point P that divides the directed segment from A to B in the ratio 2 to 1. Chelsea constructed a point Q that divides the directed segment from B to A in the ratio 1 to 2. How are points P and Q related? Explain.
Given the directed line segment from A to B, construct the point P that divides the segment in the given ratio from A to B.
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y
0-2-4
2
4
-2
-4
42
x
CCity Hall
Main LibraryM
18. City planners use a number line to place landmarks along a new street. Each unit of the number line represents 100 feet. A fountain F is located at coordinate −3 and a plaza P is located at coordinate 21. The city planners place two benches along the street at points that divide the segment from F to P in the ratios 1 to 2 and 3 to 1. What is the distance between the benches?
19. The course for a marathon includes a straight segment from city hall to the main library. The planning committee wants to put water stations along this part of the course so that the stations divide the segment into three equal parts. Find the coordinates of the points at the which the water stations should be placed.
20. Multi-Step Carlos is driving on a straight section of highway from Ashford to Lincoln. Ashford is at mile marker 433 and Lincoln is at mile marker 553. A rest stop is located along the highway 2 _ 3 of the distance from Ashford to Lincoln. Assuming Carlos drives at a constant rate of 60 miles per hour, how long will it take him to drive from Ashford to the rest stop?
21. The directed segment from J to K is shown in the figure.
Points divide the segment from J to K in the each of the following ratios. Which points have integer coordinates? Select all that apply
A. 1 to 1
B. 2 to 1
C. 2 to 3
D. 1 to 3
E. 1 to 2
Module 12 650 Lesson 2
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H.O.T. Focus on Higher Order Thinking
22. Critique Reasoning Jeffrey was given a directed line segment and was asked to use a compass and straightedge to construct the point that divides the segment in the ratio 4 to 2. He said he would have to draw a ray and then construct 6 congruent segments along the ray. Tamara said it is not necessary to construct 6 congruent segments along the ray. Do you agree? If so, explain Tamara’s shortcut. If not, explain why not.
23. Explain the Error Point A has coordinate -9 and point B has coordinate 9. A student was asked to find the coordinate of the point P that is 2 _ 3 of the distance from A to B. The student said the coordinate of point P is −3.
a. Without doing any calculations, how can you tell that the student made an error?
b. What error do you think the student made?
24. Analyze Relationships Point P divides the directed segment from A to B in the ratio 3 to 2. The coordinates of point A are (-4, -2) and the coordinates of point P are (2, 1) . Find the coordinates of point B.
25. Critical Thinking _ RS passes through R (−3, 1) and S (4, 3) . Find a point P on
_ RS
such that the ratio of RP to SP is 5 to 4. Is there more than one possibility? Explain.
panyLesson Performance TaskIn this lesson you will subdivide line segments in given ratios. The diagram shows a line segment divided into two parts in such a way that the longer part divided by the shorter part equals the entire length divided by the longer part:
a _ b
= a + b _ a
Each of these ratios is called the Golden Ratio. To find the point on a line segment that divides the segment this way, study this figure:
In the figure, LMQS is a square. LN ___ LM equals the Golden Ratio (the entire segment length divided by the longer part).
1. Describe how, starting with line segment _ LM , you can find the location of point N.
2. Letting LM equal 1, find LN ___ LM = LN ___ 1 = LN, the Golden Ratio. Describe your method.
Module 12 652 Lesson 2
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