Midpoint Theorem of Triangles(without proof)

Midpoint Theorem

Midpoint theorem gives us the relation between the mid-segments and the edges of a triangle. A mid-segment of a triangle is the segment connecting the midpoints of any two sides. 

In the figure above, MN is a mid-segment of △ABC. A triangle has three mid-segments in total. 

The Midpoint theorem of triangles gives us a couple of relations. It says that the segment(mid-segment) joining the midpoints of any two sides of a triangle is 

• parallel to the third side of the triangle, and 

• half the length of the third side. 

So in the above figure, by midpoint theorem, the length of mid-segment MN is half the length of side BC, and MN and BC are parallel line segments. So if BC is 10 units, MN is 5 units. 

In mathematical terms, this can be written as : 

In △ABC, MN is parallel to BC

And MN = \(\frac{1}{2}\) BC …..(1)

Let O be the midpoint of side BC in △ABC. 

MO and NO are the other two midsegments of △ABC along with MN. And so again, by the midpoint theorem of triangles, 

MO is parallel to AC & MO = \(\frac{1}{2}\) AC …...(2)

NO is parallel to AB & NO = \(\frac{1}{2}\) AB ……(3)

When we connect the midpoints of the sides of a given triangle we get a new triangle inside of the given triangle and it is called the medial triangle of the given triangle. △MNO is the medial triangle of △ABC.

(1) + (2) + (3) will give us,

MN + MO + NO = \(\frac{1}{2} BC + \frac{1}{2} AC + \frac{1}{2} AB \)

MN + MO + NO = \(\frac{1}{2}\) (BC + AC + AB)

So if we sum the three sides of a triangle and half that sum, the result will be equal to the sum of the sides of its medial triangle. 

Converse of Midpoint Theorem

Consider a line segment(MN) whose one end(M) is the midpoint of a side(AB) of a triangle. The other end rests somewhere on an adjacent side(AC). 

Also, the segment is parallel to the third side(BC) of the triangle. 

If these two conditions are met, the converse of the midpoint theorem guarantees that the other end of the segment is the midpoint of the side of the triangle on which it rests. So in the figure above, N is the midpoint of side AC according to the converse. 

Here’s a very mathematical looking textbook example. 

From the data given in the figure, find QC. 

P is the midpoint of AB. So one end of line segment PQ is the midpoint of a side of △ABC. And the line segment PQ is parallel to BC. 

By the converse of the midpoint theorem, the other end of the segment PQ must be the midpoint of the side AC. So Q is the midpoint of AC, which means AQ = QC. 

AC = 5 units, so QC = \(\frac{1}{2}\) AC = \(\frac{1}{2} \cdot\) 5

So QC = 2.5 units.