Join two midpoints
The midpoint of a side is the point exactly halfway along it. Mark the midpoints of two sides of a triangle and join them with a straight line. That line is called a midline.
The midpoint theorem says: the line segment joining the midpoints of two sides of a triangle is parallel to the third side and half as long.
A triangle whose base, along the bottom, is 4 units long. The line across the middle joins the midpoints of the two sloping sides.
Count it on the grid
Each sloping side goes 2 across and 4 up. Its midpoint is halfway in both directions: 1 across and 2 up from the bottom corner. Both midpoints are 2 up from the base, at the same height, so the line joining them is level, like the base. It is parallel to the base.
The midline spans 2 units across and the base spans 4. The midline is exactly half as long.
Why it is true
Look at the small triangle above the midline, and at the whole triangle. They share the top angle. The two sides of the small triangle that meet there run from the top corner to the midpoints, so each is half of a side of the whole triangle. The small triangle has two sides in the ratio 1 : 2 to the whole triangle, with the same angle between them, so by the SAS test for similarity the two triangles are similar, with scale factor .
Two things follow. The third sides are in the same ratio, 1 : 2, so the midline is half as long as the base. And the angles match: the midline meets each sloping side at the same angle as the base does. Two lines that cross a third line at equal corresponding angles are parallel, so the midline is parallel to the base.
The small triangle above the midline, shaded inside the whole triangle. Each of its sides is half of the matching side of the whole, and its angles are the same.
Any triangle
The first triangle has two equal sides, but the argument never used that, so the theorem holds for every triangle. In the triangle below, the top corner is A, the bottom left corner is B and the bottom right corner is C. No two sides are equal, and the base BC is not level: it goes 8 across and 2 down.
AB goes 2 to the left and 4 down from A, so its midpoint is 1 to the left and 2 down. AC goes 6 to the right and 6 down, so its midpoint is 3 to the right and 3 down. The midline between them goes 4 across and 1 down. That is the direction of BC, so the midline is parallel to BC, and it is half of BC in each direction, so it is half as long.
A triangle with no two sides equal. The base BC goes 8 across and 2 down, and the midline goes 4 across and 1 down.
Four triangles from three midlines
Join all three midpoints, and the triangle is cut into four smaller triangles. Each midline is half of the side it is parallel to, and each side of the big triangle is cut into two halves. So each of the four small triangles has sides that are half of the big triangle's three sides, and by SSS the four are congruent. Each one is a quarter of the whole triangle.
That matches the rule for area: a triangle at half scale has of the area. In the triangle below, the whole triangle has a base of 8 and a height of 6, so its area is ½ × 8 × 6 = 24 square units. Each small triangle has a base of 4 and a height of 3, so its area is ½ × 4 × 3 = 6, and 4 × 6 = 24.
The three midlines cut the triangle into four congruent triangles: one at each corner, and one upside down in the middle.
The usual mistakes
Halving the wrong side. The midline is parallel to, and half of, the side it does not touch. The midline joining the midpoints of AB and AC is half of BC, not half of AB.
Taking the midline to be as long as the base because it is parallel to it. Parallel lines can have any lengths; this one is half.
Giving the area ratio for the length ratio. The small top triangle has sides half as long as the whole, a ratio of 1 : 2, and a quarter of the area, a ratio of 1 : 4.
Worked example: The Braces of a Garden Swing Frame
Question The end frame of a garden swing is a triangle ABC. The legs AB and AC are each 3 m long and the feet B and C stand 3.6 m apart on level ground. A brace MN joins M, the midpoint of AB, to N, the midpoint of AC. (a) How long is the brace MN, and why is it level? (b) A second brace joins N to P, the midpoint of BC. How long is NP?
1.M is the midpoint of AB and N is the midpoint of AC. By the midpoint theorem, MN is parallel to the third side BC and MN = 12 BC.
M and N are the midpoints of AB and AC, so by the midpoint theorem MN is parallel to BC and half of it. 2.(a) MN = 12 × 3.6 = 1.8 m. The brace is level because it is parallel to BC, and BC lies along the level ground.
(a) MN = 12 × 3.6 = 1.8 m, and it is level because BC is level. 3.For the second brace, N is the midpoint of CA and P is the midpoint of CB, so the third side of the triangle is AB.
N and P are the midpoints of CA and CB, so the third side is now AB. 4.(b) NP = 12 AB = 12 × 3 = 1.5 m, and NP is parallel to the leg AB. Check: triangle AMN has sides 1.5 m, 1.5 m and 1.8 m, each exactly half of 3 m, 3 m and 3.6 m.
(b) NP = 12 × 3 = 1.5 m, parallel to the leg AB.
Answer: (a) 1.8 m; (b) 1.5 m
Common mistakes
- Halving the wrong side: taking MN to be half of AB because M lies on AB. The brace MN is parallel to and half of the side it does not touch, which is BC.
- Measuring NP as half of BC because P is the midpoint of BC. The third side for the brace NP is AB, the side that neither N nor P lies on.
More congruence, similarity and circle theorems problems, worked step by step →