Theorem

Midpoint theorem

Geometry • Triangle

Statement

The segment joining the midpoints of two sides of a triangle is parallel to the third side and equal to half of it; consequently the medial triangle formed by all three midpoints is similar to the original with ratio

\frac{1}{2}

, and so has one quarter of its area.

How to use it

1Use it to halve a length without any measurement, or to prove two lines parallel.
2Iterating it builds a sequence of nested triangles whose areas form a geometric progression with ratio $\frac{1}{4}$ .
3The converse holds too: a line through one midpoint parallel to another side bisects the third.

Worked example

△ A B C

P

and

Q

are the midpoints of

A B

and

A C

. If

B C = 14

cm, find

P Q

Step-by-step

1
$P Q = \frac{1}{2} B C$
Why: $P$ and $Q$ are midpoints of two sides, so $P Q$ is half the third.
2
$= 7$ cm

Answer

7

Watch out

The medial triangle has half the side but a quarter of the area, so counting halvings when areas are wanted is an off-by-one waiting to happen. The segment must join midpoints of two sides; a segment from a midpoint to any other point has no such property.

Builds on

Also known as

midsegment theorem, line joining midpoints, medial triangle, half the third side

Appears in past papers

Statement

Worked example

Appears in past papers

2020 (II)

2020 (I)

2019 (II)

2018 (II)

2018 (I)

2017 (II)

2017 (I)

2016 (II)