Mid Point Theorem

Last Updated : 27 Feb, 2026

The midpoint theorem states that if a segment is formed by connecting the midpoints of two of the sides of a triangle, then that segment must be parallel to the third side and equal to third side of the triangle

Given: A triangle ABC in which D is the midpoint of AB and E is the midpoint of AC.

mid_point_theorem
  • To Prove: DE ∥ BC and DE = 1/2(BC)
  • Construction: Extend the line segment joining points D and E to F such that DE = EF, and join CF.

Proof:

In ∆AED and ∆CEF:

  • DE = EF (construction)
  • ∠1 = ∠2 (vertically opposite angles)
  • AE = CE (E is the mid-point)

By SAS criterion, △AED ≅ △CEF.

Therefore, ∠3 =∠4 (by CPCT)

But these are alternate interior angles, so AB ∥ CF

  • AD = CF  (CPCT)
  • But AD = DB (D is the mid-point), so BD = CF

In Quadilateral BCFD:

  • BD∥ CF (since AB ∥ CF)
  • BD = CF

BCFD is a parallelogram as one pair of opposite sides is parallel and equal.

Therefore, 
DF∥ BC (opposite sides of parallelogram)
DF = BC (opposite sides of parallelogram)

As DF∥ BC, DE∥ BC and DF = BC

But DE = EF
So, DF = 2(DE)
2(DE) = BC
DE = 1/2(BC)

Hence, proved that the line joining the mid-points of two sides of the triangle is parallel to the third side and is half of it.

Midpoint Formula

The midpoint of any line segment is defined as the coordinate point that divides the line segment into two equal parts.

Suppose P(x1, y1) and Q(x2, y2) are the coordinates of the endpoints of any line segment; then the midpoint formula is given as

Midpoint = [\frac{(x_1 + x_2)}{2}, \frac{(y_1 + y_2)}{2}]

Converse of Midpoint Theorem Proof

The line drawn through the midpoint of one side of a triangle parallel to the base of a triangle bisects the third side of the triangle.

converse_of_mid_point_theorem
  • Given: In △PQR, S is the midpoint of PQ, and ST ∥ QR. 
  • To Prove: T is the midpoint of PR.
  • Construction: Draw a line through R parallel to PQ and extend ST to U.

Proof:

  • ST∥ QR   (given)
  • So, SU∥ QR
  • PQ∥ RU   (construction)

Therefore, Quadilateral SURQ is a parallelogram.

  • SQ = RU   (Opposite sides of parallelogram)
  • But SQ = PS   (S is the mid-point of PQ)
  • Therefore, RU = PS

In △PST and △RUT

  • ∠1 =∠2   (vertically opposite angles)
  • ∠3 =∠4   (alternate angles)
  • PS = RU  (proved above)

△PST ≅ △RUT by AAS criteria

Therefore, PT = RT

T is the mid-point of PR.

Solved Examples of Midpoint Theorem

Example 1: l, m, and n are three parallel lines. p and q are two transversals intersecting parallel lines at A, B, C, D, E, and F as shown in the figure. If AB:BC = 1:1, find the ratio of DE:EF.

3
  • Given: AB:BC = 1:1
  • To find: DE: EF
  • Construction: Join AF such that it intersects line m at G.

In △ACF 
AB = BC (1:1 ratio)
BG∥ CF (as m ∥ n)

Therefore, by converse of mid-point theorem G is the midpoint of AF (AG = GF)

Now, in △AFD

AG = GF (proved above)
GE∥ AD (as l ∥ m)

Therefore, by converse of mid-point theorem E is the mid-point of DF (FE = DE)

So, DE:EF = 1:1 (as they are equal)

Example 2: In the figure given below, L, M, and N are midpoints of sides PQ, QR, and PR, respectively, of triangle PQR.

If PQ = 8 cm, QR = 9 cm, and PR = 6 cm. Find the perimeter of the triangle formed by joining L, M, and N.

4

Solution:

As L and N are mid-points

By mid-point theorem 
LN ∥ QR and LN = 1/2 × (QR)
LN = 1/2 × 9 = 4.5cm

Similarly, LM = 1/2 × (PR) = 1/2×(6) = 3cm
Similarly, MN = 1/2 × (PQ) = 1/2 × (8) = 4cm

Therefore,
Perimeter of △LMN = LM + MN + LN
Perimeter of △LMN = 3 + 4 + 4.5 = 11.5 cm
Perimeter is 11.5cm

Practice Questions on Midpoint Theorem

Question 1: In triangle ABC, the midpoints of sides AB and AC are M and N, respectively. If MN is drawn parallel to side BC, and BC = 10 cm, what is the length of MN?

Question 2: Given triangle PQR with midpoints S and T on sides PQ and PR, respectively. Prove that segment ST is parallel to side QR and that its length is half of QR.

Question 3: In triangle DEF, the vertices are at D(2, 3), E(6, 7), and F(4, 5). Find the coordinates of the midpoints G and H of sides DE and DF, respectively, and determine the length of segment GH.

Question 4: In quadrilateral ABCD, let M and N be the midpoints of sides AB and CD, respectively. If MN is drawn, show that MN is parallel to the midpoints of sides AD and BC.

Question 5: In triangle XYZ, let X(1, 1), Y(5, 3), and Z(3, 7). Find the lengths of segments connecting the midpoints of XY, XZ, and YZ. Verify the Midpoint Theorem.

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