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)

The nearest bus stop $B$ from Tadeo's house $H$ is on a straight road leading directly to the golf course $G .$

Directed line segment HS with point B marked on the line

The distance from

H

B

is one-fifth of the distance from

H

G .

What point on the graph represents the bus stop

B ?

Answer in exact form.

We know that the distance from H to B is one-fifth of the distance from Tadeo's house to the gym. This means if we partition HG in five congruent segments, one of them will fall on HB and four will fall on BG.

To arrive at B, Tadeo has to travel one-fifth of the rise and one-fifth of the run between H and G, starting from H. Let's find these distances by using the Slope Formula.

Knowing the rise and the run, we can figure out the x- and y-coordinate of B by adding 15 of the run and the rise to the x- and y-coordinates of H respectively. These coordinates are ( -4, - 2). B_x: & - 4+1/5( 9)= - 11/5 [0.8em] B_y: & - 2+1/5( 4)= - 6/5 The bus stop is at B(- 115,- 65).

Point $C$ divides the directed line segment $\overline{X Y}$ such that the ratio of $\overline{X C}$ to $\overline{C Y}$ is $3$ to $5 .$ Point $F$ divides the segment so that the ratio of $\overline{X F}$ to $\overline{F Y}$ is $6$ to $2 .$

Where should you place a third point

P

in order for it to divide

\overline{Y X}

so that the ratio of

\overline{Y P}

\overline{P X}

5

3 ?

Both C and F partitions the directed line segment XY in 8 congruent segments. Let us show this in the diagram.

Now we want to find the point that divides the directed line segment YX in a ratio of 5 to 3. Notice how the order of Y and X has been switched. This means we are counting the number of segments starting from Y. Again, the ratio is 5 to 3. Therefore, we still need 8 congruent segments of which 5 belong to YP and 3 belong to PX.

As we can see, P would have to fall on C as well.

What are the coordinates of point $P ?$

Line segment from (2,2) to (6,4) divided into 2 congruent parts. From (6,4) the segement is extended with a further 5 congruent parts to point P.

Examining the diagram, we see that P lies beyond the directed line segment AB. We also notice that P lies on the x-coordinate 16. Therefore, we only need to find the y-coordinate of P. Let's draw two right triangles, one with AB as hypotenuse and another with AP as hypotenuse.

According to the Angle-Angle Similarity Theorem, these triangles are similar. Therefore, we can write the following equation. y_P-y_A/14=2/4 Let's solve this equation for y_P.

To get the y-coordinate of P, we have to add 7 to the y-coordinate of A. In the diagram we see that the y-coordinate of A is 2.

The coordinates of P are (16,9).

Ali is hiking in the forest. He has marked his current position on a coordinate plane as $A (3, 5)$ and he is heading towards $B (20, 15)$ along a straight line and with a constant speed. Ali will reach point $B$ in $10$ hours. What is the location $P$ of Ali at the end of the third hour? Give your answer in decimal form.

Directed line segment A(3,5) to B(20,15)

From the exercise, we know that it takes 10 hours to travel AB. Let's divide the directed line segment into 10 congruent segments, where each segment represents 1 hour of walking. We can then mark the point P, representing where Ali is after 3 hours.

To travel from A to P, we need to move 310 of the horizontal and vertical distance between A and B. Let's measure these distances.

We see that P is both to the right and above point A. Therefore, to get the coordinates of P, we must add 310 of the horizontal and the vertical distance between A and B to the x- and y-coordinates of A( 3, 5). x_P:& 3 +3/10( 17)=8.1 [1em] y_P:& 5 +3/10( 10)=8 The coordinates of P are (8.1,8).

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

Recommended exercises

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	10 Exercises - Grade E - A
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