A median halves a triangle's area

The idea

In a triangle, the line joining a vertex to the midpoint of its opposite side divides it into two triangles of equal area, as they have equal bases and the same height.

Four triangles

The two diagonals of a rectangle cut it into four triangles, numbered 1 to 4. They are not copies of each other: 1 and 3 are wide and flat, 2 and 4 are narrower and taller.

Which triangle has the greatest area?

• Triangle 1 or 3, the wide flat ones
• Triangle 2 or 4, the taller ones
• All four have the same area

Slide M, drag A

M is a point on the base BC, and the line AM cuts the triangle into two parts. Slide M and drag A to see the areas of the two parts. Make the parts equal in all three places on the list.

The line to the midpoint

In a triangle, the line joining a vertex to the midpoint of its opposite side divides it into two triangles of equal area, as they have equal bases and the same height.

This line is called a median. In the toy, ABM and AMC always have the same height: the distance from A to the line BC. So the area depends only on the base, ½ × BM × h against ½ × MC × h, and the two are equal exactly when BM = MC.

Look at A = 1 unit to the right of B, height 5, with M the midpoint. ABM and AMC are not copies of each other, because AB and AC are different lengths. Yet ½ × 4 × 5 = 10 and ½ × 4 × 5 = 10: the areas are equal.

The same fact explains the rectangle. Its diagonals bisect each other, so O is the midpoint of BD, and CO is a median of triangle BCD: triangles OBC and ODC have equal areas. Going round, every neighbouring pair is equal, so all four triangles have the same area, one quarter of the rectangle.

Notes

In a triangle, the line joining a vertex to the midpoint of its opposite side divides it into two triangles of equal area, as they have equal bases and the same height.

Check yourself

M is the midpoint of BC. Why do the triangles ABM and AMC have equal areas?

ABCD is a rectangle of area 72 cm². Its diagonals meet at O. What is the area of triangle AOB, in cm²?

Answer: 18

All four triangles have the same area, so each is 72 ÷ 4 = 18 cm².

Triangle ABC has an area of 24 cm². M is the midpoint of BC and N is the midpoint of AM. What is the area of the shaded triangle BNM, in cm²?

Answer: 6

AM is a median of ABC, so ABM = 24 ÷ 2 = 12 cm². BN is a median of ABM, so BNM = 12 ÷ 2 = 6 cm².

In triangle ABC, M is on BC with BM : MC = 1 : 3, and the area of ABC is 20 cm². What is the area of ABM?

• They are congruent: one is the mirror image of the other. Look at the picture: AB and AC are different lengths, so they are not copies. Equal areas need only equal bases and the same height.
• They have equal bases, BM = MC, and the same height from A — correct. Yes! The height is the distance from A to the line BC, the same for both, and BM = MC.
• AM is perpendicular to BC. Here AM slants, and the areas are still equal. The height is measured from A straight down to the line BC, not along AM.
• 5 cm² — correct. Yes! ABM and AMC have the same height, so their areas are in the ratio of the bases, 1 : 3. ABM is 1 part out of 4: 20 ÷ 4 = 5 cm².
• 10 cm². That is half of 20, which is true only when M is the midpoint. Here BM is shorter than MC.
• 15 cm². That is the area of AMC, the bigger part (3 parts out of 4). ABM is the smaller part.