Proportional segments: the side-splitter theorem

Warm-up

Three quick questions from earlier lessons.

Discover

A line parallel to one side of a triangle cuts the other two sides. If it cuts one side in the ratio 1 : 2, how does it cut the other?

Also $1 : 2$
It depends on the angle

Slide DE. Divide AD by DB, and AE by EC.

The two quotients are always equal
AD always equals AE

Key idea

Parallel lines split sides proportionally.

Side-splitter theorem: if

DE \parallel BC

with

D

on

AB

and

E

on

AC

, then

\dfrac{AD}{DB} = \dfrac{AE}{EC}

, and

\triangle ADE \sim \triangle ABC

. The segment joining two midpoints (a midsegment) is parallel to the third side and half as long.

Worked example

$AD = 4$ , $DB = 6$ , $AE = 6$ , $DE \parallel BC$ .

\dfrac{4}{6} = \dfrac{6}{EC}

AD/DB = AE/EC.

EC = 9

Cross-multiply.

Your turn, guided

Your turn: $BC = 18$ . The midsegment parallel to $BC$ .

9

Half.

Practice ★

In $\triangle ABC$ , a line parallel to $BC$ meets $AB$ at $D$ and $AC$ at $E$ , with $AD = 5$ , $DB = 3$ and $AE = 9$ . Find $EC$ .Not to scale
In $\triangle ABC$ , a line parallel to $BC$ meets $AB$ at $D$ and $AC$ at $E$ , with $AD = 4$ , $DB = 3$ and $AE = 9$ . Find $EC$ .Not to scale

In a triangle, $BC = 26$ . The segment joining the midpoints of $AB$ and $AC$ has length…

Watch out!

Practice ★★

$DE \parallel BC$ with $D$ on $AB$ , $E$ on $AC$ ; $AD = 2$ , $DB = 7$ , $AE = 3$ . Find $AC$ .Not to scale

Boss problem. In $\triangle ABC$ , $DE \parallel BC$ with $D$ on $AB$ , $AD = 6$ , $DB = 1$ and $DE = 8$ .

Exit ticket

Three questions to check you have got it.

In $\triangle ABC$ , a line parallel to $BC$ meets $AB$ at $D$ and $AC$ at $E$ , with $AD = 5$ , $DB = 3$ and $AE = 9$ . Find $EC$ .Not to scale
In $\triangle ABC$ , a line parallel to $BC$ meets $AB$ at $D$ and $AC$ at $E$ , with $AD = 4$ , $DB = 3$ and $AE = 9$ . Find $EC$ .Not to scale
In $\triangle ABC$ , a line parallel to $BC$ meets $AB$ at $D$ and $AC$ at $E$ , with $AD = 7$ , $DB = 8$ and $AE = 5$ . Find $EC$ .Not to scale

Summary card

Proportional segments

$DE \parallel BC$ : $AD/DB = AE/EC$ ; midsegment = half the third side.

Watch out: keep the parts in matching order.

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