Mastering the Directed Line Segment Formula and Partitioning Line Segments

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.

What are the coordinates of the point at which Dylan would have walked

3

miles?

Example

Finding Points on a Triangle's Perimeter

In $△ P Q R,$ point $S$ partitions $\overline{P Q}$ from point $P$ to point $Q$ in the ratio $3 : 7 .$ Similarly, point $T$ partitions $\overline{R P}$ 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 $T .$

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 $\overline{P Q}$ and $\overline{R P} .$

Next, the scale factor

k

can be found by using the formula for the scale factor of a dilation.

k = \frac{3}{3 + 7} \Leftrightarrow k = 0.3

Now, the coordinates of

S

can be found by substituting the coordinates of the endpoints of

\overline{P Q}

into the formula for the point that partitions the directed line segment. Note that

S

divides

\overline{P Q}

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

$a (- b) = - a \cdot b$

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
$\overline{P Q}$	$P (- 2, 5)$ 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 $P (- 2, 5)$	$0.3$	$T (5 + 0.3 (- 2 - 5), 1 + 0.3 (5 - 1))$	$T (2.9, 2.2)$

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.

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 .

\frac{3}{1 . 5} \Leftrightarrow 2 : 1

With this information, the scale factor of the dilation can be determined.

k = \frac{2}{2 + 1} \Leftrightarrow k = \frac{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 + \frac{2}{3} (6.5 - 2), 5 + \frac{2}{3} (2 - 5))

Evaluate

SubTerms

Subtract terms

D (2 + \frac{2}{3} (4.5), 5 + \frac{2}{3} (- 3))

MultPosNeg

$a (- b) = - a \cdot b$

D (2 + \frac{2}{3} (4.5), 5 - \frac{2}{3} (3))

MoveRightFacToNum

$\frac{a}{c} \cdot b = \frac{a \cdot b}{c}$

D (2 + \frac{9}{3}, 5 - \frac{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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Hint

Solution

Hint

Solution