Partitioning a segment

Warm-up

Three quick questions from earlier lessons.

Discover

From $A(0, 0)$ to $B(6, 3)$ . Where is the point one third of the way along?

$(2, 1)$
$(3, 1.5)$

Key idea

Move a fraction of the way, or take a weighted average.

The point a fraction

t

of the way from

A

to

B

is

A + t(B - A)

. A point dividing

AB

in the ratio

m : n

(from

A

) is

\left(\dfrac{n x_A + m x_B}{m + n}, \dfrac{n y_A + m y_B}{m + n}\right)

, a weighted average.

Worked example

$A(1, 2)$ , $B(9, 6)$ , ratio $1 : 3$ .

t = \tfrac{1}{4}

1 part of 4.

P = (3, 3)

¼ of (8, 4) = (2, 1).

Your turn, guided

Your turn: x-coordinate halfway between 2 and 10.

6

Average.

Practice ★

Tap the point one third of the way from $A(-3, -2)$ to $B(3, 4)$ .
Tap the point one third of the way from $A(-5, -6)$ to $B(4, 3)$ .

Point $P$ divides segment $AB$ in the ratio $1 : 2$ (closer to $A$ ). $A$ has $x = -4$ and $B$ has $x = 2$ . Find the $x$ -coordinate of $P$ .

Watch out!

Practice ★★

Point $P$ divides $AB$ in the ratio $3 : 1$ from $A$ . $A$ has $x = -2$ , $B$ has $x = 1$ . Find $P$ 's $x$ -coordinate.
Point $P$ divides $AB$ in the ratio $1 : 4$ from $A$ . $A$ has $x = -2$ , $B$ has $x = 2$ . Find $P$ 's $x$ -coordinate.

Exit ticket

Three questions to check you have got it.

Tap the point one third of the way from $A(-3, -2)$ to $B(3, 4)$ .
Tap the point one third of the way from $A(-5, -6)$ to $B(4, 3)$ .
Tap the point one third of the way from $A(-7, -2)$ to $B(2, 1)$ .

Summary card

Partitioning a segment

$A + t(B - A)$ ; ratio $m : n$ gives $t = \dfrac{m}{m + n}$ .

Watch out: count the parts of the ratio.

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