Directed Segments

Try this

Drag only $B$ , from the right of $A$ over to its left. When does the sign of the value written next to arrow $A B$ change?
Arrange the points in the order $A$ , $C$ , $B$ (with $C$ in the middle). Now $B C$ goes backwards. Is $A B + B C$ still equal to $A C$ ?
Three points can sit on the line in $6$ different orders. Try all of them. Does the light ever go out?
Make $A$ and $C$ coincide. What is $A B + B C$ ? How would you say that in terms of “going there and coming back”?

On a line with a chosen positive direction (a directed line, or axis), a segment with a chosen start and end is called a directed segment, written $A B$ . Its length together with a sign showing its direction is called its signed value, written $A B$ ; its length is written $∣ A B ∣$ . If it points the same way as the axis, the value is positive; if it points the opposite way, negative. So

A B = - B A, A B + B A = 0, ∣ A B ∣ = ∣ B A ∣.

And on a number line there is a rule that is “always right”:

A B = x_{B} - x_{A} (end coordinate minus start coordinate)

Predict first

On a number line, $A$ , $B$ , $C$ have coordinates $4$ , $- 1$ , $6$ . Without drawing, what is $A B + B C$ ?

Choose an answer first; then you can see the result.

Why: Chasles's relation

In plane geometry, “ $A B + B C = A C$ ” only holds when $B$ lies between $A$ and $C$ . With directions, it holds for every order. This is the addition theorem for directed segments, also called Chasles's relation.

Proof: AB + BC = AC for any three points on a line

Step 1 / 5

Start with the easiest case
$A$ , $B$ , $C$ in order from left to right: all three segments point along the axis, so their values are just their lengths. Then $A B + B C = A C$ simply says “the parts add up to the whole”.

$5 + 2 = 7$
The key step: value = end coordinate − start coordinate
Add, and the middle term cancels
Test another order: B on the far left
Generalize: a chain of segments joined end to start

Ratios of values: λ

Signed values can be divided. If $P$ is a point on the line $P_{1} P_{2}$ , the ratio

λ = \frac{P _{1} P}{P P _{2}}

is called the ratio in which $P$ divides the directed segment $P_{1} P_{2}$ . Mind the order: from the start $P_{1}$ to the dividing point $P$ , then from $P$ to the end $P_{2}$ . When $P$ lies between $P_{1}$ and $P_{2}$ , the two pieces point the same way and $λ > 0$ : $P$ is an internal dividing point. When $P$ lies outside, the pieces point opposite ways and $λ < 0$ : an external dividing point.

Drag

P

, or set

λ

directly. On the

λ

scale below, the triangle never reaches the value marked by the red circle.

Solving $λ = \frac{x - x _{1}}{x _{2} - x}$ for $x$ :

x = \frac{x _{1} + λ x _{2}}{1 + λ} (λ \neq = - 1)

Predict first

Keep $P_{1}$ and $P_{2}$ fixed. Let $P$ run to the right along the axis, very, very far. Which number does $λ$ get closer and closer to?

0

1

- 1

It keeps growing without end

Choose an answer first; then you can see the result.

Remove the direction

Break it

Counterexample lab: remove one condition

Conclusion: $A B + B C = A C$ for any three points

The directed version: drag anything you like, the light is always ✓.

Practice

Exercise 1Basic

On a number line, $A$ and $B$ have coordinates $- 3$ and $4$ . Find the signed value $B A$ .

Exercise 2Basic

Four points $A$ , $B$ , $C$ , $D$ lie on a number line with $A B = 3$ , $B C = - 7$ , $C D = 2$ . Find $D A$ .

Exercise 3Advanced

On a number line, $P_{1}$ and $P_{2}$ have coordinates $- 1$ and $5$ . The point $P$ divides $P_{1} P_{2}$ in the ratio $λ = - 3$ . Find the coordinate of $P$ .

Exercise 4Challenge

On a number line, $P_{1}$ and $P_{2}$ have coordinates $0$ and $6$ . Find the coordinates of all points $P$ with $∣ P_{1} P ∣ = 2∣ P P_{2} ∣$ . (Separate them with commas.)

Try this

Why: Chasles's relation

Proof: AB + BC = AC for any three points on a line

Ratios of values: λ

Remove the direction

Counterexample lab: remove one condition

Practice

Mistake Museum · Exhibits from this stop

Taking the signed value for the length

Thinking λ can be any real number