Theorem

A median bisects the area of a triangle

Geometry • Triangle • Medians and Centroid

Formula

[A B D] = [A C D] = \frac{1}{2} [A B C]

Write

[X Y Z]

for the area of triangle

X Y Z

. If

A D

is the median of triangle

A B C

to the side

B C

— so

D

is the midpoint of

B C

— then

[A B D] = [A C D] = \frac{1}{2} [A B C]

. The reason is that

B D = D C

while both triangles have the same height, the perpendicular from

A

to the line

B C

; more generally, two triangles with equal bases on one line and a common apex have equal areas, so the same conclusion holds with the apex at any point of the line

A D

How to use it

1When a midpoint appears on a side, read it as an area bisection at once — no lengths or heights need to be computed.
2Apply it repeatedly: a median of one of the two halves quarters the original area, and so on.
3Take the apex at any point $P$ of the median $A D$ to get $[P B D] = [P C D]$ ; subtracting that pair from $[A B D] = [A C D]$ shows $[P A B] = [P A C]$ .
4Combine it with the converse reading: if a cevian from $A$ bisects the area, its foot is the midpoint of $B C$ .

Worked example

In triangle

A B C

the area is

48 cm^{2}

A D

is the median to

B C

, and

E

is the midpoint of

A D

. Find the area of triangle

A B E

Step-by-step

1
$[A B D] = \frac{1}{2} [A B C] = \frac{1}{2} \times 48 = 24 cm^{2}$
Why: $A D$ is a median of triangle $A B C$ .
2
$[A B E] = \frac{1}{2} [A B D] = 12 cm^{2}$
Why: In triangle $A B D$ , the segment $B E$ is the median to $A D$ , so it bisects that area in turn.

Answer

12 cm^{2}

Watch out

It is the median that bisects the area, not any cevian: a segment from

A

to a point

D^{'}

B C

splits the area in the ratio

B D^{'} : D^{'} C

, so only the midpoint gives halves. Equal areas do not mean congruent triangles — the two halves generally have different perimeters — and

A D

bisects neither the angle

A

nor the side

A B

unless

A B = A C

Builds on

Area of a triangle from base and height

Also known as

median bisects area, median divides a triangle into two equal areas, equal bases and equal heights give equal areas, median area bisection

Appears in past papers

2020 (II)

Q85