Mastering the Directed Line Segment Formula and Partitioning Line Segments

Formula: $(x_{1} + k (x_{2} - x_{1}), y_{1} + k (y_{2} - y_{1}))$
Segment	Endpoints	Scale Factor	Substitute	Simplify
$\overline{P Q}$	$P (- 2, 5)$ and $Q (- 4, 1)$	$0.3$	$S (- 2 + 0.3 (- 4 - (- 2)), 5 + 0.3 (1 - 5))$	$S (- 2.6, 3.8)$
$\overline{R P}$	$R (5, 1)$ and $P (- 2, 5)$	$0.3$	$T (5 + 0.3 (- 2 - 5), 1 + 0.3 (5 - 1))$	$T (2.9, 2.2)$

Formula:

(x_{1} + k (x_{2} - x_{1}), y_{1} + k (y_{2} - y_{1}))

Segment

Endpoints

Scale Factor

Substitute

Simplify

\overline{P Q}

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)

\overline{R P}

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)

City	Longitude ( $x$ )	Latitude ( $y$ )
Madrid	$\approx - 3.70$	$\approx 40.42$
Warsaw	$\approx 21.01$	$\approx 52.23$

City

Longitude (

x

)

Latitude (

y

)

Madrid

\approx - 3.70

\approx 40.42

Warsaw

\approx 21.01

\approx 52.23

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 $(- 3.70, 40.42)$	$0.25$	$M (21.01 + 0.25 (- 3.70 - 21.01), 52.23 + 0.25 (40.42 - 52.23))$	$M (14.83, 49.28)$

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)

City	Longitude ( $x$ )	Latitude ( $y$ )
Houston	$\approx - 95.37$	$\approx 29.76$
Washington D.C.	$\approx - 77.04$	$\approx 38.91$

City

Longitude (

x

)

Latitude (

y

)

Houston

\approx - 95.37

\approx 29.76

Washington D.C.

\approx - 77.04

\approx 38.91

Station	Ratio	Scale Factor	Simplify
$A$	$1 : 4$	$k = \frac{1}{1 + 4}$	$k = 0.2$
$B$	$2 : 3$	$k = \frac{2}{2 + 3}$	$k = 0.4$
$C$	$3 : 2$	$k = \frac{3}{3 + 2}$	$k = 0.6$
$D$	$4 : 1$	$k = \frac{4}{4 + 1}$	$k = 0.8$

Station

Ratio

Scale Factor

Simplify

A

1 : 4

k = \frac{1}{1 + 4}

k = 0.2

B

2 : 3

k = \frac{2}{2 + 3}

k = 0.4

C

3 : 2

k = \frac{3}{3 + 2}

k = 0.6

D

4 : 1

k = \frac{4}{4 + 1}

k = 0.8

Formula: $(x_{1} + k (x_{2} - x_{1}), y_{1} + k (y_{2} - y_{1}))$
Station	Scale Factor	Substitute	Simplify
$A$	$0.2$	$A (- 95.37 + 0.2 (- 77.04 - (- 95.37)), 29.76 + 0.2 (38.91 - 29.76))$	$A (- 91.70, 31.59)$
$B$	$0.4$	$B (- 95.37 + 0.4 (- 77.04 - (- 95.37)), 29.76 + 0.4 (38.91 - 29.76))$	$B (- 88.04, 33.42)$
$C$	$0.6$	$C (- 95.37 + 0.6 (- 77.04 - (- 95.37)), 29.76 + 0.6 (38.91 - 29.76))$	$C (- 84.37, 35.25)$
$D$	$0.8$	$D (- 95.37 + 0.8 (- 77.04 - (- 95.37)), 29.76 + 0.8 (38.91 - 29.76))$	$D (- 80.71, 37.08)$

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)

Find the coordinates of point $P$ along the directed line segment $\overline{A B}$ such that $\overline{A P}$ to $\overline{P B}$ is in the ratio of $4 : 1 .$ Answer in exact form.

Directed line segment AB with a point P in between A and B

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).

Find the coordinates of point $P$ along the directed line segment $\overline{A B}$ such that $\overline{A P}$ to $\overline{P B}$ is in the ratio $2 : 3 .$ Answer in exact form.

Diagram with segment from B(-2,-4) to A(6,1). Point P along the line is marked.

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).

What is the ratio of $\overline{N L}$ to $\overline{N M} ?$

A directed line segment NM with a point L in between N and M.

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.

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.

Diagram with train station at the origin, the factory in (2,2), and the circus in (5,5).

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.

Position of Points on a Line Segment

Catch-Up and Review

Investigating a Partitioned Line Segment

Partitioning a Directed Line Segment

Hint

Solution

General Method for Partitioning a Directed Line Segment

Finding Points on a Triangle's Perimeter

Hint

Solution

Partitioning Distances Between Cities on a Map

Hint

Solution

Partitioning a Distance Into Congruent Segments

Hint

Solution

Finding a Point on a Directed Line Segment

Hint

Solution

Position of Points on a Line Segment

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